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A similarity transformation is utilized to reduce the generalized nonlinear Schrödinger (NLS) equation with variable coefficients to the standard NLS equation with constant coefficients, whose rogue wave solutions are then transformed back into the solutions of the original equation. In this way, Ma breathers, the first- and second-order rogue wave solutions of the generalized equation, are constructed. Properties of a few specific solutions and controllability of their characteristics are discussed. The results obtained may raise the possibility of performing relevant experiments and achieving potential applications.

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... It should be noted that the models describing these physical systems are most often based on the nonlinear Schrödinger (NLS) equation and its variants, with the rogue waves representing the rational solutions [6]. In addition, other mathematical models such as the NLS equation with variable coefficients [7], the derivative NLS equation [8], Hirota's equation [9,10], Davey-Stewartson equation [11], and other, have also been shown to possess rogue wave solutions. By studying rogue waves in different fields, two common characteristics have been observed: the wave amplitude is at least twice the amplitude of surrounding waves and the localization occurs in both the temporal and spatial domains. ...

... The present paper is written to explore this management venue theoretically and numerically. In Sec. 2, we extend our previous similarity method, given in [7,32], to reduce the generalized NLS equation with variable coefficients and an external potential to the standard NLS equation. In Sec. 3, we analyze the formation of several types of optical rogue waves and their interactions, by selecting special nonlinearity coefficients, including the first, second and third-order managed rogue waves. ...

... , 2, j = ), where j s correspond to the eigenvalues in the Darboux method [44].Their results show that for the standard first-order rogue waves there are no parameters j s , the general second-order rogue wave has one parameter 1 s , while the general third-order rogue wave has two parameters 1 s and 2 s , and so on[43]. A similar procedure is devised in [44] as well.To connect the rogue wave solution of Eq. (1) with those of Eq.(2), we use the similarity transformation[7, 32] ...

Using a similarity transformation, we obtain analytical solutions to a class of nonlinear Schrödinger (NLS) equations with variable coefficients in inhomogeneous Kerr media, which are related to the optical rogue waves of the standard NLS equation. We discuss the dynamics of such optical rogue waves via nonlinearity management, i.e., by selecting the appropriate nonlinearity coefficients and integration constants, and presenting the solutions. In addition, we investigate higher-order rogue waves by suitably adjusting the nonlinearity coefficient and the rogue wave parameters, which could help in realizing complex but controllable optical rogue waves in properly engineered fibers and other photonic materials. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement.

... Periodic and hyperbolic wave functions may display the dynamical behavior of roguelike wave phenomena. The profiles of the first-order and second-order rogue wave solutions of the inhomogeneous NLSE with variable coefficients can be controlled by a number of parameters [24]. A common characteristic of all these waves is that they ride on a finite background. ...

... Hence, the external potential is just a simple quadratic potential, modulated by the diffraction coefficient. In this manner, we try to stay close to the physically relevant situations, so when (1), we have obtained bright and dark soliton solutions by means of the F-expansion method in [25]; the first-order and the second-order rogue waves were also obtained, and the dynamical behavior of those waves was discussed in our previous work [24]. However, the important controllable behavior of rogue waves in [24] has not been investigated, not even for N(z,x) = N (x), and also the third-order rogue waves have not been analyzed at all. ...

... In this manner, we try to stay close to the physically relevant situations, so when (1), we have obtained bright and dark soliton solutions by means of the F-expansion method in [25]; the first-order and the second-order rogue waves were also obtained, and the dynamical behavior of those waves was discussed in our previous work [24]. However, the important controllable behavior of rogue waves in [24] has not been investigated, not even for N(z,x) = N (x), and also the third-order rogue waves have not been analyzed at all. We focus in this paper on spatially localized solutions for which N(z,x) = N (x). ...

We demonstrate controllable parabolic-cylinder optical rogue waves in certain inhomogeneous media. An analytical rogue wave solution of the generalized nonlinear Schrödinger equation with spatially modulated coefficients and an external potential in the form of modulated quadratic potential is obtained by the similarity transformation. Numerical simulations are performed for comparison with the analytical solutions and to confirm the stability of the rogue wave solution obtained. These optical rogue waves are built by the products of parabolic-cylinder functions and the basic rogue wave solution of the standard nonlinear Schrödinger equation. Such rogue waves may appear in different forms, as the hump and paw profiles.

... Similarity transformation has extensively been used to construct analytical solutions of the Vc-NLSE connecting the solutions of constant coefficient NLSE [40]. Over the years utilizing this method, several theoretical studies have been reported on rogue wave dynamics [41][42][43][44][45][46]. Also, the study of breathers and rational solutions of different Vc-NLSE have revealed several new features, which include nonlinear tunnelling effect in periodically distributed system and exponentially dispersion decreasing fiber [47] and Peregrine comb as multiple compression point in the amplitude in periodically modulated fibers [48]. ...

... Recently it has been reported that periodic modulation of nonlinearity coefficient along the transverse axis leads to evolution of Akhmediev-like breathers [44]. Also, a previous report revealed how KM breathers and first and second order rational solutions evolve under periodic modulation of both dispersion and nonlinearity coefficients [42]. Zhong et al. have shown that under such condition, KM breathers propagate in a periodically modulated background with three peaks in one breathing unit, while the rational solutions maintain their standard features. ...

... Also, the parameters β 2 (z) and χ(z) are the group velocity dispersion (GVD) and the nonlinear (self-phase modulation) coefficients, respectively. Motivated by previous studies [42][43][44], we utilize the well-known similarity transformation in order to solve Eq. (1), which is written as ...

Optical rogue waves and its variants have been studied quite extensively in the context of optical fiber in recent years. It has been realized that dispersion management in optical fiber is experimentally much more feasible compared to its nonlinear counterpart. In this work, we report Kuznetsov-Ma (KM)-like breathers from the first three orders of rational solutions of the nonlinear Schr\"{o}dinger equation with periodic modulation of the dispersion coefficient along the fiber axis. The breather dynamics are then controlled by proper choice of modulating parameters. Additionally, the evolution of new one-peak and two-peak breather-like solutions has been displayed corresponding to the second-order rational solution. Direct numerical simulations based on modulational instability has also been executed which agree well with the analytical results, thereby making the proposed system more feasible for experimental realization.

... In 2000, Ferman et al. began to explore the formation and propagation characteristics of self-similar pulses in fiber amplifiers [11]; in 2003, Finot et al. looked for the evolution of self-similar pulses in fiber amplifiers and the interaction between two adjacent pulses [12]; in 2000, Soljacic et al. [13] used all-optical devices to propose the application of self-similar pulses in optical fiber communication systems. We also have employed the self-similar method to study rogue waves [14,15]. In a word, the self-similar method consists in transforming a NL equation which is difficult to solve into a NL equation which can be solved by a coordinate transformation. ...

... In a word, the self-similar method consists in transforming a NL equation which is difficult to solve into a NL equation which can be solved by a coordinate transformation. This novel research method has gained wide acceptance in the twenty-first century and has stimulated more interest in the selfsimilar NL wave packets [11][12][13][14][15]. ...

An effective and simple method to solve nonlinear evolution partial differential equations is the self-similarity transformation, in which one utilizes solutions of the known equation to find solutions of the unknown. In this paper, we employ an improved similarity transformation to transform the \((2+1)\)-dimensional (D) sine-Gordon (SG) equation into the \((1+1)\)-D SG equation and obtain non-rational solutions of the \((2+1)\)-D SG equation by utilizing the known solutions of the \((1+1)\)-D SG equation. Based on the solutions obtained, and with the help of special choices of the involved solution parameters, several localized structures of the \((2+1)\)-D SG model are analyzed on a finite background, such as the embedded hourglass, split silo, dumbbell, and pie solitons. Their spatiotemporal profiles are displayed, and their properties are discussed.

... In particular, rogue wave solutions emerging from optical fibers have been found analytically for many types of generalized NLS models such as NLS models with constant coefficients [10,[26][27][28][29] and NLS models with varying coefficients [9,30,31]. Recently, this interesting phenomenon of optical rogue waves has been verified experimentally [21,32]. ...

... According to previous works [9,30], we consider the above symmetry (reduction) transformation or similarity transformation (4) that would reduce Eq. (3) to the standard NLS equation ...

We consider the inhomogeneous nonparaxial nonlinear Schr\"odinger (NLS) equation with varying dispersion, nonlinearity, and nonparaxiality coefficients, which governs the nonlinear wave propagation in an inhomogeneous optical fiber system. We present the similarity and Darboux transformations and for the chosen specific set of parameters and free functions, the first- and second-order rational solutions of the nonparaxial NLS equation are generated. In particular, the features of rogue waves throughout polynomial and Jacobian elliptic functions are analyzed, showing the nonparaxial effects. It is shown that the nonparaxiality increases the intensity of rogue waves by increasing the length and reducing the width simultaneously, by the way it increases their speed and penalizes interactions between them. These properties and the characteristic controllability of the nonparaxial rogue waves may give another opportunity to perform experimental realizations and potential applications in optical fibers.

... From this preliminary method, varying coefficients are obtained but the complex field is deduced from the modified Darboux transformation or from the Lax pair method [55]. In what follows, we use the envelope field in the form [52,56,57] ...

... According to the previous works [52,57], we use the symmetry transformation given by Eq. (13) that would reduce Eq. (12) to the integrable Hirota equation in the form [58] ...

We derive the nonlinear Schrödinger (NLS) equation in chiral optical fiber with right- and left-hand nonlinear polarization. We use the similarity transformation to reduce the generalized chiral NLS equation to the higher-order integrable Hirota equation. We present the first- and second-order rational solutions of the chiral NLS equation with variable and constant coefficients, based on the modified Darboux transformation method. For some specific set of parameters, the features of chiral optical rogue waves are analyzed from analytical results, showing the influence of optical activity on waves. We also generate the exact solutions of the two-component coupled nonlinear Schrödinger equations, which describe optical activity effects on the propagation of rogue waves, and their properties in linear and nonlinear coupling cases are investigated. The condition of modulation instability of the background reveals the existence of vector rogue waves and the number of stable and unstable branches. Controllability of chiral optical rogue waves is examined by numerical simulations and may bring potential applications in optical fibers and in many other physical systems.

... For example, Serkin presented a systematic method to construct analytical solutions of the one-dimensional nonlinear Schrödinger equation with variable coefficients and discussed nonautonomous solitary wave dispersion management [16]. Recently, we have helped in the development and use of a number of methods, such as the homogeneous balance principle and Fexpansion technique [17,18], self-similar transformation [19][20][21][22], and Hirota's bilinear method [23,24]. Very recently, new solution methods appeared, such as the transformed rational function method [25], the multiple exp-function method [26], and the generalized bilinear method [27], which have introduced powerful approaches for finding meaningful solutions to various nonlinear equations. ...

... by introducing two self-similar variables X = X(x) and T = T (t) that each depend on space or time only. In our previous work, we have used a similar approach to deal with the nonlinear Schrödinger equation with variable coefficients [19,20] and obtained some interesting results. Substituting expression (2) into Eq. ...

New solitary and extended wave solutions of the generalized sinh-Gordon (SHG) equation with a variable coefficient are found by utilizing the self-similar transformation between this equation and the standard SHG equation. Two arbitrary self-similar functions are included in the known solutions of the standard SHG equation, to obtain exact solutions of the generalized SHG equation with a specific variable coefficient. Our results demonstrate that the solitary and extended waves of the variable-coefficient SHG equation can be manipulated and controlled by a proper selection of the two arbitrary self-similar functions.

... More recently, controllable behaviors of rogue waves and the related breathers have been studied [10][11][12][13][14][15][16][17][18]. The control for rogue waves [14] and superposed breather [15] were discussed. ...

A (3+1)-dimensional coupled nonlinear Schrödinger equation with different inhomogeneous diffractions and dispersion is investigated, and rogue wave and combined breather solutions are constructed. Different diffractions and dispersion of medium lead to the repeatedly excited behaviors of rogue wave and combined breather in the dispersion/diffraction decreasing system. These repeated behaviors including complete excitation, rear excitation, peak excitation and initial excitation are discussed.

... Many powerful methods to seek exact solutions to the NPDEs have been proposed in literature. Among these methods are Darboux transformation [1], Hirota bilinear method [2], Lie group method [3], homogeneous balance method [4], F-expansion technique [5], similarity transformation method [6], exp-Function method [7], sine-cosine method [8], tanh-sech method [9] and extended tanh-coth method [10] and so on. ...

In this paper we present an improvement to those methods based on the using of elementary functions like exponential, trigonometric and hyperbolic functions in obtaining exact solutions to nonlinear partial differential equations (NPDEs). The improved method is applied to stable nonlinear Schrödinger (NLS), unstable NLS, generalized NLS, High-order NLS and derivative NLS equations. New solutions for these equations are obtained. The obtained solutions are more general than a wide class of previous solutions. Solutions of a derivative NLS equation that describes the large-amplitude solitons propagating in an arbitrary direction in a high- hall plasma are also obtained. Moreover, the method is applied to magnetohydrodynamics (MHD) equations describing an ideal incompressible flow in the steady state. One of the most important advantages of the solution method presented here is it deals with several types of nonlinearities associated with PDEs without making a transformation of the original equation to another one.

... Many powerful methods to seek exact solutions to the NPDEs have been proposed in literature. Among these methods are Darboux transformation [1], Hirota bilinear method [2], Lie group method [3], homogeneous balance method [4], F-expansion technique [5], similarity transformation method [6], exp-Function method [7], sine?cosine method [8], tanh?sech method [9] and extended tanh?coth method [10] and so on. ...

... To solve this problem, we use the symmetry reduction method [56,57] to obtain some integrabil- ity conditions and to reduce the generalized nonparaxial chiral NLS equation to the higher-order integrable Hirota equation. So doing, we use the envelope field in the form [55,58,59] ...

The generalized nonparaxial nonlinear Schrödinger (NLS) equation in optical fibers filled with chiral materials is reduced to the higher-order integrable Hirota equation. Based on the modified Darboux transformation method, the nonparaxial chiral optical rogue waves are constructed from the scalar model with modulated coefficients. We show that the parameters of nonparaxiality, third-order dispersion, and differential gain or loss term are the main keys to control the amplitude, linear, and nonlinear effects in the model. Moreover, the influence of nonparaxiality, optical activity, and walk-off effect are also evidenced under the defocusing and focusing regimes of the vector nonparaxial NLS equations with constant and modulated coefficients. Through an algorithm scheme of wider applicability on nonparaxial beam propagation methods, the most influential effect and the simultaneous controllability of combined effects are underlined, showing their properties and their potential applications in optical fibers and in a variety of complex dynamical systems.

... For a single component NLSE with variable dispersion and nonlinearity but with linear gain or loss, a similarity variable transformation can be applied to reduce the problem to one with homogeneous properties. Deducing rogue wave modes then becomes feasible [7]. For the case of a potential trap, one relevant physical situation is Bose-Einstein condensate and the corresponding analytical description is then the Gross-Pitaevskii equation [8]. ...

In this paper, the effect of gain or loss on the dynamics of rogue waves is investigated by using the complex Ginzburg-Landau equation as a framework. Several external energy input mechanisms are studied, namely, constant background or compact Gaussian gains and a ‘rogue gain’ localized in space and time. For linear background gain, the rogue wave does not decay back to the mean level but evolves into peaks with growing amplitude. However, if such gain is concentrated locally, a pinned mode with constant amplitude could replace the time transient rogue wave and become a sustained feature. By restricting such spatially localized gain to be effective only for a finite time interval, a ‘rogue-wave-like’ mode can be recovered. On the other hand, if the dissipation is enhanced in the localized region, the formation of rogue wave can be suppressed. Finally, the effects of linear and cubic gain are compared. If the strength of the cubic gain is large enough, the rogue wave may grow indefinitely (‘blow up’), whereas the solution under a linear gain is always finite. In conclusion, the generation and dynamics of rogue waves critically depend on the precise forms of the external gain or loss.

... In mathematical and soliton theory, Ref. [12] has already reported the first rogue wave solution in the nonlinear Scrödinger equation in 1983. Later, there were presented abundant rogue wave of various of nonlinear Schrödinger equations [13][14][15][16]. ...

By means of Hirota bilinear method and symbolic computation, this paper studied two types of interaction solutions for the potential Yu–Toda–Sasa–Fukuyama equation. The first type of interaction between lump soliton and one stripe soliton generate the fusion and fission phenomenon, while the second type of interaction solution between lump soliton and twin stripe solitons generate the rogue wave. The corresponding dynamic plots of the above phenomena are graphically displayed respectively.

... Scalar systems are governed by the nonlinear Schrodinger equation (NLSE), and analytically solutions of rogue waves are essential for predicting their dynamics. Scalar analytical solutions were developed by Peregrine [18] based on the work of Ma [19] and later by Akhmediev [20][21][22] and are known as rational solutions [23][24][25][26][27][28][29][30][31]. Other methods were introduced, such as the bi-linear method [32], and the Darboux transformation [27,33], which gave similar results. ...

Rogue waves, which were first described as an oceanographic phenomenon, constitute an important factor in the dynamics of many physical systems. Most of these systems were analyzed by scalar fields, while some of them, and specifically in optics, are described by a vector field. Thus, they differ from scalar systems in several crucial aspects. In this work, we study experimentally twin-peak rogue waves with a temporal imaging system capable of measuring the Stokes vectors as a function of time. We found that the two peaks in optical twin-peak rogue waves have orthogonal states of polarization and similar intensities. We observed this with two different systems, however, we do not have a theoretical explanation for this phenomena and could not explain it with current models.

... Here we investigate their occurrence and management in BEC for the choice of parameters which are experimentally feasible. We would like to bring to notice that several works have been reported that deal with the management of rogue waves obtained via similarity transformation in different contexts [33][34][35][36][37]. In nonlinear optics, their controllable behaviour is depicted by taking different functional form of dispersion parameters, corresponding to different nonlinear fiber systems [34,38]. ...

We present the coherent control of rogue wave triplets in Bose–Einstein condensate (BEC) for the parametric choices which support their occurrence experimentally. This analysis has been done for two cases involving the trapping of BEC in (i) time-independent (ii) time-dependent harmonic potential. We have found that these waves no longer lie on the apex of the equilateral triangle as in the case of homogeneous system but can be manipulated by modulating the trap. This enables us to control the relative distance among the three rogue wave peaks and their amplitude, for a given nonlinearity parameter. These results can be helpful to stimulate the study of rogue wave triplets experimentally in BEC.

... Searching for exact solutions to NLEEs makes contributions to understanding the corresponding physical system, especially when numerical solutions fail [16][17][18][19][20][21][22][23][24]. Various exact solutions, including soliton solutions [25,26], breather wave solutions [27][28][29][30], lump solutions [31][32][33][34][35][36][37], and rogue wave solutions [38][39][40][41] have been derived in many ways. Soliton solutions are a kind of traveling wave solutions, which are generated based on the balance of nonlinear effects and dispersion in the medium [42]. ...

The N-rational solutions to two (2+1)-dimensional nonlinear evolution equations are constructed by utilizing the long wave limit method. M-lump solutions to the two equations are derived by making some parameters conjugate to each other. We present and discuss the 1-, 2- and 3-lump solutions to the two equations. The amplitude and shape of the one lump wave remain unchanged during the propagation. The dynamic properties of the collisions among multiple lump waves are analyzed, which indicate that the fusion and fission of multiple lump waves might occur. The multiple lump waves might merge into one lump wave, then split into multiple lump waves. The lines which multiple lump waves follow are various if we choose different parameters. These results are helpful to describe some nonlinear phenomena in the areas of optics, fluid dynamics and plasma.

... Aiming at the NLSE, the nonlinear wave solution (including several basic wave solutions such as soliton, Kuznetsov-Ma breather, Akhmediev breather, W-shaped soliton, rogue wave, period wave and so on) can be obtained by Darboux transformation method [17][18][19][20][21]. The influence factors of nonlinear waves are analyzed by linear stability method, and the corresponding relationship between nonlinear wave and MI in gain distribution plane phase diagram can be constructed. ...

We discuss modulation instability for the generalized nonlinear Schrödinger equation based on nonzero background wave frequency. First of all, we analyze the existence condition of modulation instability under different perturbed frequency. The influences of the background amplitude, background frequency and perturbed frequency on the modulation instability gain are researched, respectively. Also, we obtain the correspondences between several nonlinear excitations (Kuznetsov-Ma breather, general breather, rogue wave, bright soliton and plane wave) and modulation instability according to new parameters. Furthermore, by the Fourier transformation method, we perform spectrum analyses of the first-order and second-order rogue waves. The perturbed frequency of the rogue wave can affect the location and profile of the spectrum. And we find that the spectrum of the second-order rogue wave is jagged due to the collision of the rogue waves. These results would help us further understand the dynamics of rogue wave in complex systems.

... Many physical quantities are related to the time variables; for instance, the viscosity is a function of timedependent temperature [36]. Nonlinear partial differential equations with variable coefficients can describe more realistic phenomenon than those with constant coefficients [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] if we consider the media and boundary inhomogeneities. Such models can characterize the nonlinear waves, for example, the solitons, rogue waves and breathers evolution in inhomogeneous media [54]. ...

In this paper, we study the (3+1)-dimensional variable-coefficient Kudryashov–Sinelshchikov (vc-KS) equation, which characterizes the evolution of nonautonomous nonlinear waves in bubbly liquids. The nonautonomous lump solutions of the vc-KS equation are produced via the Hirota bilinear technique. The characteristics of trajectory and velocity of this wave are analyzed with variable dispersion coefficients. Based on the positive quadratic function assumption, we further discuss two types of interactions between the soliton and lump under the periodic and exponential modulations. Then, we give the breathing lump waves showing the periodic oscillation behavior. Finally, we obtain the second-order nonautonomous lump solution, which also shows periodic interactions if we select trigonometric functions as the dispersion coefficients.

... In addition, controllable rogue waves have also been extensively studied. For example, the management of rogue waves in inhomogeneous nonlinear media was reported in [22][23][24][25]. In these papers only the dispersion (diffraction) and nonlinearity management were explored, which are usually considered as functions of the propagation distance [26][27][28][29]. ...

Nonlinear Schrödinger equation with simple quadratic potential modulated by a spatially-varying diffraction coefficient is investigated theoretically. Second-order rogue wave breather solutions of the model are constructed by using the similarity transformation. A modal quantum number is introduced, useful for classifying and controlling the solutions. From the solutions obtained, the behavior of second order Kuznetsov-Ma breathers (KMBs), Akhmediev breathers (ABs), and Peregrine solitons is analyzed in particular, by selecting different modulation frequencies and quantum modal parameter. We show how to generate interesting second order breathers and related hybrid rogue waves. The emergence of true rogue waves - single giant waves that are generated in the interaction of KMBs, ABs, and Peregrine solitons - is explicitly displayed in our analytical solutions.

We study the (3+1)-dimensional nonlinear Schrödinger equation with different distributed transverse diffraction and dispersion based on the similarity transformation and obtain exact spatiotemporal breather solutions. Based on these solutions, two kinds of combined Akhmediev breather and Kuznetsov-Ma soliton display their controllable properties by modulating the values of the maximum effective distance \(Z_\mathrm{m}\) and the effective distance at peak of breather \(Z_0\) . Our results indicate that the relation between \(Z_\mathrm{m}\) and \(Z_0\) is the basis to realize the control and manipulation of propagation behaviors of breathers, such as complete excitation, sustainment, restraint and recurrence. These results are potentially useful for future experiments in various blood vessels, optical communications, and long-haul telecommunication networks.

We analytically study optical rogue waves in the presence of quintic nonlinearity and nonlinear dispersion effects. Dynamics of the rogue waves are investigated through the precise expressions of their peak, valley, trajectory, and width. Based on this, the properties of a few specific rogue waves are demonstrated in detail, and the dynamical evolution of rogue waves can be well controlled under different nonlinearity management. It shows that the peak reaches its maximum and the valley becomes minimized when the width evolves to the minimum value. Moreover, we find that the higher-order effects here achieve balance due to the integrability, and they only influence the rogue waves' trajectory.

In this paper, a unified formula of a series of rogue wave solutions for the standard (1+1)-dimensional nonlinear Schrödinger equation is obtained through exp-function method. Further, by means of an appropriate transformation and previously obtained solutions, rogue wave solutions of the variable coefficient Schrödinger equation are also obtained. Two free functions of time t and several arbitrary parameters are involved to generate a large number of wave structures.

Under investigation in this paper is a amplifier nonlinear Schrödinger Maxwell-Bloch (NLS-MB) system which describes the propagation of optical pulses in an inhomogeneous erbium doped fiber. Nonautonomous breather and rogue wave (RW) solutions of the amplifier NLS-MB system are constructed via the modified Darboux transformation with the inhomogeneous parameters. By suitably choosing the dispersion coefficient function, several types of inhomogeneous nonlinear waves are obtained in: (1) periodically fluctuating dispersion profile; (2) exponentially increasing (or decreasing) dispersion profile; and (3) linearly decreasing (increasing) dispersion profile. The nonautonomous characteristics of the breathers and RWs are graphically investigated, including the breather accelerating and decelerating motions, boomerang breather, breather compression, breather evolution, periodic RW, boomerang RW and stationary RW. Such novel patterns as the periodic breathers and rogue-wave fission of the amplifier NLS-MB system are exhibited by properly adjusting the group velocity dispersion function and interaction parameter between silica and doped atoms.

We study the amplitude modulation of ion-acoustic wave (IAW) packets in an
unmagnetized electron-ion plasma with two-temperature (cool and hot) electrons
in the context of the Tsallis' nonextensive statistics. Using the
multiple-scale technique, a nonlinear Schr{\"o}dinger (NLS) equation is derived
which governs the dynamics of modulated wave packets. It is shown that in
nonextensive plasmas, the IAW envelope is always stable for long-wavelength
modes $(k\rightarrow0)$ and unstable for short-wavelengths with $k \gtrsim1$.
However, the envelope can be unstable at an intermediate scale of perturbations
with $0<k<1$. Thus, the modulated IAW packets can propagate in the form of
bright envelope solitons or rogons (at small- and medium scale perturbations)
as well as dark envelope solitons (at large scale). The stable and unstable
regions are obtained for different values of temperature and density ratios as
well as the nonextensive parameters $q_c$ and $q_h$ for cool and hot electrons.
It is found that the more (less) the population of superthermal cool (hot)
electrons, the smaller is the growth rate of instability with cutoffs at
smaller wave numbers of modulation.

We investigate a (2+1)-dimensional-coupled variable coefficient nonlinear Schrödinger equation in parity time symmetric nonlinear couplers with gain and loss and analytically obtain a combined structure solution via the Darboux transformation method. When the imaginary part of the eigenvalue \(n\) is smaller or bigger than 1, we can obtain the combined Peregrine soliton and Akhmediev breather, or Kuznetsov–Ma soliton, respectively. Moreover, we study the controllable behaviors of this combined Peregrine soliton and Kuznetsov–Ma soliton structure in a diffraction decreasing system with exponential profile. In this system, the effective propagation distance \(Z\) exists a maximal value \(Z_m\) and the maximum amplitude of the KM soliton appears in the periodic locations \(Z_{i}\) . By modulating the relation between values of \(Z_m\) and \(Z_i\) , we realize the control for the excitation of the combined Peregrine soliton and Kuznetsov–Ma soliton, such as the restraint, maintenance, and postpone.

In this paper, we study the families of solitary-wave solutions to the inhomogeneous coupled nonlinear Schrödinger equations with space- and time-modulated coefficients and source terms. By means of the similarity reduction method and Möbius transformations, many types of novel temporal solitary-wave solutions of this nonlinear dynamical system are analytically found under some constraint conditions, such as the bright-bright, bright-dark, dark-dark, periodic-periodic, W-shaped, and rational wave solutions. In particular, we find that the localized rational-type solutions can exhibit both bright-bright and bright-dark wave profiles by choosing different families of free parameters. Moreover, we analyze the relationships among the group-velocity dispersion profiles, gain or loss distributions, external potentials, and inhomogeneous source profiles, which provide the necessary constraint conditions to control the emerging wave dynamics. Finally, a series of numerical simulations are performed to show the robustness to propagation of some of the analytically obtained solitary-wave solutions. The vast class of exact solutions of inhomogeneous coupled nonlinear Schrödinger equations with source terms might be used in the study of the soliton structures in twin-core optical fibers and two-component Bose-Einstein condensates.

We introduce two-dimensional (2D) linear and nonlinear Talbot effects (LTE
and NLTE). They are produced by 2D diffraction patterns and can be visualized
as 3D stacks of Talbot carpets. The NLTE originates from 2D rogue waves and
forms in a bulk 3D nonlinear medium. The recurrence of an input rogue wave
(with a $\pi$ phase shift) can only be observed at the (half) Talbot length.
Different from NLTE, the LTE displays the usual fractional Talbot images as
well. We also find that the smaller the period of incident rogue waves, the
shorter the Talbot length. Increasing the beam intensity increases the Talbot
length, but above a threshold this leads to a catastrophic self-focusing
phenomenon which destroys the nonlinear effect.

We present the angular vector soliton solutions of the coupled (2+1)dimensional nonlinear Schrödinger (NLS) equations via a similarity transformation that is connected with the stationary NLS equation. Then we investigate the transverse spatial distributions of the controllable vector soliton clusters. We obtain exact angular vector soliton solutions that are constructed with the help of Whittaker special functions. We find that these solitons can be effectively controlled by the modulation depth, the topological charge, and the radial quantum number. Our results show that, for integer or fractional topological charge, the intensity profiles of these vector solitons exhibit various forms, such as the vortex-ring shapes and either symmetric or asymmetric necklace-ring patterns.

We investigate optical temporal rogue waves of the generalized nonlinear Schrödinger (NLS) equation with higher-order odd and even terms and space- and time-modulated coefficients, which includes the NLS equation, Lakshmanan-Porsezian-Daniel (LPD) equation, Hirota equation, Chen-Lee-Liu equation, and Kaup-Newell derivative NLS equation. Based on the similarity reduction method, the generalized NLS equation can be reduced to the integrable LPD-Hirota equation under a set of constraint conditions, from which the solutions of generalized NLS equation can be obtained in the basis of solutions of the LPD-Hirota equation and the similarity transformation. In particular, the first- and second-order self-similar rogue wave solutions of the generalized NLS equation are derived under different parameters, and the contour profiles and density evolutions of self-similar rogue wave solutions are given to study their wave structures and dynamic properties. At the same time, the motions of the hump and valleys related to the self-similar rogue waves are also given, from which we can control and manage the self-similar rogue waves by adjusting the third-order dispersion term. These results may be useful in nonlinear optics and related fields.

Exact solutions of nonlinear models might explain real physical, engineering and biophysical problems, including blood pressure waves, -helical protein, matter wave and hardening spring effect, and so on. We re-study the breaking soliton model and obtain eleven kinds of variable separation solution by means of the improved tanh-function method with three different ansätz, that is, positive-power ansatz, radical sign combined ansatz, and positive and negative power-symmetric ansatz. By careful analysis of these exact solutions, we find that these seemly independent variable separation solutions actually depend on each other. Therefore, different ansätz of improved tanh-function method are equivalent.

By applying the similarity transformation we find analytical solutions of the generalized nonlinear Schrödinger equation with spatially modulated parameters and a special external potential in the form of periodic breathing nonlinear wave packets. These breathers are built by the products of parabolic-cylinder functions and the one-soliton solutions with the continuous-wave background or the second-order soliton solutions of the standard nonlinear Schrödinger equation with constant coefficients. Some examples of such composed solutions possessing applicative potential are given, in which the breathers appear in different forms. The numerical simulations are performed to compare the numerical solution with the analytical solution and to confirm the stability of the breathers found. © 2014, Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg.

In the present work, the Fokas system that acts an important role in the monomode optical fibers is considered. The Cole-Hopf transformation is employed to construct the condition for finding the diverse soliton solutions via introducing different trial functions. Some new soliton solutions such as the M-shaped rational, interaction, cross-periodic, double exponential form and multi-waves soliton solutions are obtained. The dynamic behaviors of the solutions are described through the 3-D plot, 2-D contour and 2-D curve. The ideas presented in this work can be applied to explore the other PDEs arising in the optics.

We construct rogue waves (RWs) in a coupled two-mode system with the
self-focusing nonlinearity of the Manakov type (equal SPM and XPM
coefficients), spatially modulated coefficients, and a specially designed
external potential. The system may be realized in nonlinear optics and
Bose-Einstein condensates. By means of a similarity transformation, we
establish a connection between solutions of the coupled Manakov system with
spatially-variable coefficients and the basic Manakov model with constant
coefficients. Exact solutions in the form of two-component Peregrine and
dromion waves are obtained. The RW dynamics is analyzed for different choices
of parameters in the underlying parameter space. Different classes of RW
solutions are categorized by means of a naturally introduced control parameter
which takes integer values.

The dynamics of the modulation rogue waves in an inhomogeneous nonlinear optical fiber with the periodic modulation are studied. We find that for different modulation amplitudes and modulation frequencies, the modulation rogue wave solution can be Peregrine comb, rogue wave, or the transition state in between, respectively. In particular, the phase diagram of the three kinds of nonlinear states is given at the modulation amplitude and modulation frequency plane. Moreover, the dynamics characteristics of the Peregrine comb and the rogue wave are discussed on the localized soliton background. It is interesting that the main excitation characteristics of the Peregrine combs and the rogue waves on an infinitely wide plane wave background are well maintained on the soliton background. These results pave the way for exciting and manipulating the rogue waves on a local background.

This paper investigates new rogue wave solutions of the nonlinear Schrödinger equation with variable coefficients, utilizing the self-similar transformation method. A new rogue wave family is introduced, which displays different wave structures. When one chooses appropriate variable coefficients, a series of first-order, second-order, and third-order rogue waves are obtained. We explore the generation and interaction of rational polynomial rogue waves in detail, and discuss some characteristics of their structures. The paper provides theoretical basis for studying the dynamic behavior of optical rogue wave in inhomogeneous media, and presents potential application value for control of rogue waves in fibers, plasmas and other fields of physics.

A quantum simulator environment is considered in Bose-Einstein condensate under bichromatic optical lattice, manifesting rogue waves. We report an exact analytical model to produce rogue wave excitations, the dynamics of which is investigated by solving the corresponding scalar Gross-Pitaevskii equation. A qualitative understanding of the formation mechanism at various trap parameters is provided by assuming the rogue wave excitation as a mixture of bright and dark solitary waves, which offers a novel illustration of the previous conjecture for rogue waves in Bose-Einstein condensate. We identify the lattice parameters for controlling the spatio-temporal variations of the condensate density. An Anderson-like localization is identified where the condensate is seen to preserve the rogue wave nature at the central frustrated site for larger lattice frustration.

This paper studies exact solutions for the (2+1)-dimensional nonlinear Schrödinger equation with variable coefficients. First, after a transformation, the linear equation is attribute to the Hukuhara equation and then it is solved by the similarity construction method. Then, the bright and dark soliton and exact solutions for the above nonlinear equation are obtained. Finally, the graphs of these exact solutions to the linear and nonlinear equations with k=1,2 are drew, respectively, which reveals that the essential profile of solutions to the nonlinear Schrödinger equation with variable coefficients is determined by the corresponding linear equation.

The aim of this work is to study different types of similariton solutions for the inhomogeneous nonlinear Schrödinger equation (NLSE) with dual power-law nonlinearity. This is accomplished via three methods: the hyperbolic ansatz method, the auxiliary equation method, and the Riccati sub-ODE method. Depending on the Galilean change of the frame parameter Ω, the graphs reveal the fact that the rational soliton is actually a bisoliton that behaves at some aspects as a double-soliton whereas the kink boomerang-like soliton appears to be a kink-antikink soliton. These observations predict that the two sides of a rational soliton could possess distinct chirality. Furthermore, the evaluation of the modulation instability (MI) shows that the gain decreases with increasing power index p.

A wide class of nonlinear excitations and the dynamics of wave groups of finite amplitude ion-acoustic waves are investigated in multicomponent magnetized plasma system comprising warm ions, and superthermal electrons as well as positrons in presence of negatively charged impurities or dust particles. Employing the reductive perturbation technique (RPT), the Korteweg–de-Vries (KdV) equation, and extended KdV equation are derived. The presence of excess superthermal electrons as well as positrons and other plasma parameters are shown to influence the characteristics of both compressive and rarefactive solitons as well as double layers (DLs). Also, we extend our investigation by deriving the nonlinear Schrödinger equation from the extended KdV equation employing a suitable transformation to study the wave group dynamics for long waves. The analytical and numerical simulation results demonstrate that nonlinear wave predicts solitons, “table-top” solitons, DLs, bipolar structure, rogue waves, and breather structures. Moreover, implementing the concept of dynamical systems, phase portraits of nonlinear periodic, homoclinic trajectories, and supernonlinear periodic trajectories are presented through numerical simulation.

Optical rogue waves and its variants have been studied quite extensively in the context of optical fiber in recent years. It has been realized that dispersion management in optical fiber is experimentally much more feasible compared to its nonlinear counterpart. In this work, we report Kuznetsov-Ma (KM)-like breathers from the first three orders of rational solutions of the nonlinear Schrödinger equation with periodic modulation of the dispersion coefficient along the fiber axis. The breather dynamics are then controlled by proper choice of modulating parameters. Additionally, the evolution of new one-peak and two-peak breather-like solutions has been displayed corresponding to the second-order rational solution. Direct numerical simulations based on modulational instability has also been executed which agree well with the analytical results, thereby making the proposed system more feasible for experimental realization.

The research of rogue wave solutions of the nonlinear Schrödinger (NLS) equations is still an open topic. NLS equations have received particular attention for describing nonlinear waves in optical fibres, photonics, plasmas, Bose–Einstein condensates and deep ocean. This work deals with rogue wave solutions of the chiral NLS equation. We introduce an inhomogeneous one-dimensional version, and using the similarity transformation and direct ansatz, we solve the equation in the presence of dispersive and nonlinear coupling which are modulated in time and space. As a result, we show how a simple choice of some free functions can display a lot of interesting rogue wave structures and the interaction of quantum rogue waves. The results obtained may give the possibility of conducting relevant experiments in quantum mechanics and achieving potential applications.

We study the coupled nonlinear Schrödinger equation in the (2+1)-dimensional inhomogeneous (Formula presented.)-symmetric nonlinear couplers and obtain (Formula presented.)-symmetric and (Formula presented.)-antisymmetric vortex soliton solutions. The dynamical behaviors of the completely localized structures (vortex solitons) are analytically and numerically investigated in a diffraction decreasing system with exponential profile. Numerical results indicate that one vortex soliton with different topological charges can stably propagate a long distance. The space between two humps and the modulation depth of vortex solitons add when the topological charge increases. However, the change tendency of the amplitude and width of vortex solitons is same with the increase in topological charge.

One- and two-soliton analytical solutions of a fifth-order nonlinear Schrödinger equation with variable coefficients are derived by means of the Hirota bilinear method in this paper. Various scenarios of one-soliton shaping and two-soliton interaction and reshaping are investigated, using the obtained exact solutions and adjusting parameters of the underlying model. We find that widths of two colliding solitons can change without changing their amplitudes. Furthermore, we produce a solution in which two originally bound solitons are separated and are then moving in opposite directions. We also show that two colliding solitons can fuse to form a spatiotemporal train, composed of equally separated identical pulses. Moreover, we display that the width and propagation direction of the spatiotemporal train can change simultaneously. Effects of corresponding parameters on the one-soliton shaping and two-soliton interaction are discussed. Results of this paper may be beneficial to the application of optical self-routing, switching and path control.

We theoretically modulate vector soliton molecules in an optical fiber system model, when input orthogonal modes’ central wavelengths are the same or different. In the case of same central wavelength with 1064 nm, output orthogonal modes maintain their states in time or frequency domain, when projection angle or phase difference changes. While when orthogonal components have slightly different central wavelengths (2 nm wavelength difference), energy flow occurs when projection angle varies. Our results are applicable to polarization-multiplexed optical fiber communication, ultrafast optical fiber laser and so on.

We study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with higher-order effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multipeak solitons, antidark soliton, periodic wave, and W-shaped soliton. In particular, the transition condition requiring the group velocity dispersion (GVD) and third-order dispersion (TOD) to scale linearly is obtained analytically. We display several kinds of elastic interactions between the transformed nonlinear waves. We discuss the dispersion management of the multipeak soliton, which indicates that the GVD coefficient controls the number of peaks of the wave while the TOD coefficient has compression effect. The gain or loss has influence on the amplitudes of the multipeak soliton. We further derive the breather multiple births and Peregrine combs by using multiple compression points of Akhmediev breathers and Peregrine rogue waves in optical fiber systems with periodic GVD modulation. In particular, we demonstrate that the Peregrine comb can be converted into a Peregrine wall by the proper choice of the amplitude of the periodic GVD modulation. The Peregrine wall can be seen as an intermediate state between rogue waves and W-shaped solitons. We finally find that the modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues. Our results could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.

We study on non-linear localized waves on continuous wave background in a dispersion and non-linearity management fibre. We find a stable supercontinuum pulse can be generated from a small modulation on continuous wave in a proper management way, for which the pulse spectrum width and its growth rate can be controlled well by the management parameters. Additionally, we demonstrate a Kuznetsov–Ma breather like non-linear localized wave can exist in a periodic dispersion management fibre, and its spectrum evolution is distinctive from the Kuznetsov–Ma breather’s.

In terms of Darboux transformation, we have exactly solved the higher-order nonlinear Schrödinger equation that describes the propagation of ultrashort optical pulses in optical fibers. We discuss the modulation instability (MI) process in detail and find that the higher-order term has no effect on the MI condition. Under different conditions, we obtain Kuznetsov-Ma soliton and Akhmediev breather solutions of higher-order nonlinear Schrödinger equation. The former describes the propagation of a bright pulse on a continuous wave background in the presence of higher-order effects and the soliton's peak position is shifted owing to the presence of a nonvanishing background, while the latter implies the modulation instability process that can be used in practice to produce a train of ultrashort optical soliton pulses.

We study rogue waves in an inhomogeneous nonlinear optical fiber with variable coefficients. An exact rogue wave solution that describes rogue wave excitation and modulation on a bright soliton pulse is obtained. Special properties of rogue waves on the bright soliton, such as the trajectory and spectrum, are analyzed in detail. In particular, our analytical results suggest a way of sustaining the peak shape of rogue waves on the soliton background by choosing an appropriate dispersion parameter.

We study exact soliton solutions of the Sine-Gordon (SG) equation with variable coefficient. Based on the similarity transformation and Hirota's bilinear method, we report both kink-type and anti-kink-type one-soliton and two-soliton solutions of the SG equation. In particular, we analyze the emerging soliton structures by a special selection of two self-similar variables. Our results show that the shapes of both kink-type and anti-kink-type solitons can be effectively controlled by the specific form of those two self-similar variables.

We propose a scheme to demonstrate the existence of optical Peregrine rogue waves and Akhmediev and Kuznetsov-Ma breathers and realize their active control via electromagnetically induced transparency (EIT). The system we suggest is a cold, Λ-type three-level atomic gas interacting with a probe and a control laser fields and working under EIT condition. We show that, based on EIT with an incoherent optical pumping, which can be used to cancel optical absorption, (1+1)-dimensional optical Peregrine rogue waves, Akhmediev breathers, and Kuznetsov-Ma breathers can be generated with very low light power. In addition, we demonstrate that the Akhmediev and Kuznetsov-Ma breathers in (2+1)-dimensions obtained can be actively manipulated by using an external magnetic field. As a result, these breathers can display trajectory deflections and bypass obstacles during propagation.

The methodology based on the quasi-soliton concept provides for a systematic way to discover an infinite number of the novel stable bright and dark soliton management regimes for the nonlinear cubic-quintic Schrodinger equation model with varying dispersion, nonlinearity and gain or absorption. Quasi-soliton solutions for this model must be of rather general character than canonical solitons of standard nonlinear Schrodinger equation model, because the generalized model takes into account the saturation nonlinear effect and arbitrary variations of group velocity dispersion, nonlinearity and gain or absorption. Novel topological and nontopological quasi-soliton solutions for the nonlinear cubic-quintic Schrodinger equation model have been discovered. It is shown that, today, the most attractive media to discover novel topological quasi-solitons are organic thin films and polymeric waveguides.

We consider the evolution of nonlinear optical pulses in cubic-quintic nonlinear media wherein the pulse propagation is governed by the generalized nonlinear Schrödinger equation with exponentially varying dispersion, cubic, and quintic nonlinearities and gain and/or loss. Using a self-similar analysis, we find the chirped bright soliton solutions in the anomalous and normal dispersion regimes. From a stability analysis, we show that the soliton in the anomalous dispersion regime is stable, whereas the soliton in the normal dispersion regime is unstable. Numerical simulation results show that competing cubic-quintic nonlinearities stabilize the chirped soliton pulse propagation against perturbations in the initial soliton pulse parameters. We characterize the quality of the compressed pulse by determining the pedestal energy generated and compression factor when the initial pulse is perturbed from the soliton solutions. Finally, we study the possibility of rapid compression of Townes solitons by the collapse phenomenon and the exponentially decreasing dispersion. We find that the collapse could be postponed if the dispersion increases exponentially.

Solitary waves have consistently captured the imagination of scientists, ranging from fundamental breakthroughs in spectroscopy and metrology enabled by supercontinuum light, to gap solitons for dispersionless slow-light, and discrete spatial solitons in lattices, amongst others. Recent progress in strong-field atomic physics include impressive demonstrations of attosecond pulses and high-harmonic generation via photoionization of free-electrons in gases at extreme intensities of 10(14) W/cm(2). Here we report the first phase-resolved observations of femtosecond optical solitons in a semiconductor microchip, with multiphoton ionization at picojoule energies and 10(10) W/cm(2) intensities. The dramatic nonlinearity leads to picojoule observations of free-electron-induced blue-shift at 10(16) cm(-3) carrier densities and self-chirped femtosecond soliton acceleration. Furthermore, we evidence the time-gated dynamics of soliton splitting on-chip, and the suppression of soliton recurrence due to fast free-electron dynamics. These observations in the highly dispersive slow-light media reveal a rich set of physics governing ultralow-power nonlinear photon-plasma dynamics.

The Peregrine soliton is a localized nonlinear structure predicted to exist over 25 years ago, but not so far experimentally observed in any physical system. It is of fundamental significance because it is localized in both time and space, and because it defines the limit of a wide class of solutions to the nonlinear Schrödinger equation (NLSE). Here, we use an analytic description of NLSE breather propagation to implement experiments in optical fibre generating femtosecond pulses with strong temporal and spatial localization, and near-ideal temporal Peregrine soliton characteristics. In showing that Peregrine soliton characteristics appear with initial conditions that do not correspond to the mathematical ideal, our results may impact widely on studies of hydrodynamic wave instabilities where the Peregrine soliton is considered a freak-wave prototype

Some breather type solutions of the NLS equation have been suggested by Henderson et al (to appear in Wave Motion) as models for a class of 'freak' wave events seen in 2D-simulations on surface gravity waves. In this paper we first take a closer look on these simple solutions and compare them with some of the simulation data (Henderson et al to appear in Wave Motion). Our findings tend to strengthen the idea of Henderson et al. Especially the Ma breather and the so called Peregrine solution may provide useful and simple analytical models for 'freak' wave events.

The formation of breathers as prototypes of freak waves is studied within the framework of the classic ‘focussing’ nonlinear
Schrödinger (NLS) equation. The analysis is confined to evolution of localised initial perturbations upon an otherwise uniform
wave train. For a breather to emerge out of an initial hump, a certain integral over the hump, which we refer to as the “area”,
should exceed a certain critical value. It is shown that the breathers produced by the critical and slightly supercritical
initial perturbations are described by the Peregrine soliton which represents a spatially localised breather with only one
oscillation in time and thus captures the main feature of freak waves: a propensity to appear out of nowhere and disappear
without trace. The maximal amplitude of the Peregrine soliton equals exactly three times the amplitude of the unperturbed
uniform wave train. It is found that, independently of the proximity to criticality, all small-amplitude supercritical humps
generate the Peregrine solitons to leading order. Since the criticality condition requires the spatial scale of the initially
small perturbation to be very large (inversely proportional to the square root of the smallness of the hump magnitude), this
allows one to predict a priori whether a freak wave could develop judging just by the presence/absence of the corresponding
scales in the initial conditions. If a freak wave does develop, it will be most likely the Peregrine soliton with the peak
amplitude close to three times the background level. Hence, within the framework of the one-dimensional NLS equation the Peregrine
soliton describes the most likely freak-wave patterns. The prospects of applying the findings to real-world freak waves are
also discussed.
KeywordsBreathers-Nonlinear Schrödinger equation-Pulses in optical fibres-Rogue waves-Water waves

Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schrödinger equations with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general theory and use it to calculate explicitly nontrivial solutions such as periodic (breathers), resonant, or quasiperiodically oscillating solitons. Some implications to the field of matter waves are also discussed.

We present experimental and numerical results showing the generation and breakup of the Peregrine soliton in standard telecommunications fiber. The impact of nonideal initial conditions is studied through direct cutback measurements of the longitudinal evolution of the emerging soliton dynamics and is shown to be associated with the splitting of the Peregrine soliton into two subpulses, with each subpulse itself exhibiting Peregrine soliton characteristics. Experimental results are in good agreement with simulations.

The appearance of rogue waves is well known in oceanographics, optics, and
cold matter systems. Here we show a possibility for the existence of
atmospheric rogue waves.

A unidirectional optical oscillator is built by using a liquid crystal light valve that couples a pump beam with the modes of a nearly spherical cavity. For sufficiently high pump intensity, the cavity field presents complex spatiotemporal dynamics, accompanied by the emission of extreme waves and large deviations from the Gaussian statistics. We identify a mechanism of spatial symmetry breaking, due to a hypercycle-type amplification through the nonlocal coupling of the cavity field.

Recent observations show that the probability of encountering an extremely large rogue wave in the open ocean is much larger than expected from ordinary wave-amplitude statistics. Although considerable effort has been directed towards understanding the physics behind these mysterious and potentially destructive events, the complete picture remains uncertain. Furthermore, rogue waves have not yet been observed in other physical systems. Here, we introduce the concept of optical rogue waves, a counterpart of the infamous rare water waves. Using a new real-time detection technique, we study a system that exposes extremely steep, large waves as rare outcomes from an almost identically prepared initial population of waves. Specifically, we report the observation of rogue waves in an optical system, based on a microstructured optical fibre, near the threshold of soliton-fission supercontinuum generation--a noise-sensitive nonlinear process in which extremely broadband radiation is generated from a narrowband input. We model the generation of these rogue waves using the generalized nonlinear Schrödinger equation and demonstrate that they arise infrequently from initially smooth pulses owing to power transfer seeded by a small noise perturbation.

We study solitary wave solutions of the higher order nonlinear Schrodinger equation for the propagation of short light pulses in an optical fiber. Using a scaling transformation we reduce the equation to a two-parameter canonical form. Solitary wave (1-soliton) solutions exist provided easily met inequality constraints on the parameters in the equation are satisfied. Conditions for the existence of N-soliton solutions (N>1) are determined; when these conditions are met the equation becomes the modified KdV equation. A proper subset of these conditions meet the Painleve plausibility conditions for integrability.

The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach. Comment: revtex, 69 pages, 38 eps figures, new version with more references, new figures, various changes and corrections, for publ. in Rev. Mod. Phys., available also at http://www-phys.science.unitn.it/bec/BEC.html

An improved self-similar transformation is used to construct exact so-lutions of the nonlinear Schrödinger equation with variable nonlinearity and quadratic external potential, which both depend on the distance of propagation and the transverse spatial coordinate. By means of analytical and numerical methods we reveal the main features of the spatial soli-tons found. We focus on the most important optical examples, where the applied optical field is a function of both linearly or periodically varying distance and spatial coordinate. In the case of periodically varying nonlin-earity, the variations of confining external potential are found to be sign-reversible (periodically attractive and repulsive) and thus supporting the soliton management.

A detailed analysis is given to the solution of the cubic Schrödinger equation iqt + qxx + 2|q|2q = 0 under the boundary conditions as |x|∞. The inverse-scattering technique is used, and the asymptotic state is a series of solitons. However, there is no soliton whose amplitude is stationary in time. Each soliton has a definite velocity and “pulsates” in time with a definite period. The interaction of two solitons is considered, and a possible extension to the perturbed periodic wave [q(x + T,t) = q(x,t) as |x|∞] is discussed.

In 1882 Darboux proposed a systematic algebraic approach to the solution of the linear Sturm-Liouville problem. In this book, the authors develop Darboux's idea to solve linear and nonlinear partial differential equations arising in soliton theory: the non-stationary linear Schrodinger equation, Korteweg-de Vries and Kadomtsev-Petviashvili equations, the Davey Stewartson system, Sine-Gordon and nonlinear Schrodinger equations 1+1 and 2+1 Toda lattice equations, and many others. By using the Darboux transformation, the authors construct and examine the asymptotic behaviour of multisoliton solutions interacting with an arbitrary background. In particular, the approach is useful in systems where an analysis based on the inverse scattering transform is more difficult. The approach involves rather elementary tools of analysis and linear algebra so that it will be useful not only for experimentalists and specialists in soliton theory, but also for beginners with a grasp of these subjects.

We consider the family of 2nd order rogue wave rational solutions of the nonlinear Schrödinger equation (NLSE) with two free parameters. Surprisingly, these solutions describe a formation consisting of 3 separate first order rogue waves, rather than just two. We show that the 3 components of the triplet are located on an equilateral triangle, thus maintaining a certain symmetry in the solution, even in its decomposed form. The two free parameters of the family define the size and orientation of the triangle on the (x,t) plane.

We have studied Laguerre-Gaussian spatial solitary waves in strongly nonlocal nonlinear media analytically and numerically. An exact analytical solution of two-dimensional self-similar waves is obtained. Furthermore, a family of different spatial solitary waves has been found. It is interesting that the spatial soliton profile and its width remain unchanged with increasing propagation distance. The theoretical predictions may give new insights into low-energetic spatial soliton transmission with high fidelity.

We propose initial conditions that could facilitate the excitation of rogue waves. Understanding the initial conditions that foster rogue waves could be useful both in attempts to avoid them by seafarers and in generating highly energetic pulses in optical fibers.

Properties of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation (ICQNLSE) with an external potential are studied. When it is associated with the homogeneous CQNLSE, a general condition exists linking the external potential and inhomogeneous cubic and quintic (ICQ) nonlinearities. Besides for the nonpresence of an external potential, two classes of Jacobian elliptic periodic potentials are discussed in detail, and the corresponding ICQ nonlinearities are found to be either periodic or localized. Exact analytical soliton solutions in these cases are presented, such as the bright, dark, kink, and periodic solitons, etc. An appealing aspect is that the ICQNLSE can support bound states with any number of solitons when the ICQ nonlinearities are localized and an external potential is either applied or not.

An improved homogeneous balance principle and an F-expansion technique are used to construct exact periodic wave solutions to the generalized two-dimensional nonlinear Schrödinger equation with distributed dispersion, nonlinearity, and gain coefficients. For limiting parameters, these periodic wave solutions acquire the form of localized spatial solitons. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and gain (or loss). We establish a simple procedure to select different classes of solutions, using the dispersion and the gain coefficient in one case, or the chirp function and the gain coefficient in the other case, as independent parameter functions. We present a few characteristic examples of periodic wave and soliton solutions with physical relevance.

We demonstrate the existence of localized optical vortex and necklace
solitons in three-dimensional (3D) highly nonlocal nonlinear media, both
analytically and numerically. The 3D solitons are constructed with the
help of Kummer’s functions in spherical coordinates and their
unique properties are discussed. The procedure we follow offers ways for
generation, control, and manipulation of spatial solitons.

Exact N‐envelope‐soliton solutions have been obtained for the following nonlinear wave equation, i∂ψ∕∂t + i3α∣ψ∣2 ∂ψ∕∂x + β∂2ψ∕∂x2 + iγ∂3ψ∕∂x3 + δ∣ψ∣2ψ = 0, where α, β, γ and δ are real positive constants with the relation αβ = γδ. In one limit of α = γ = 0, the equation reduces to the nonlinear Schrödinger equation which describes a plane self‐focusing and one‐dimensional self‐modulation of waves in nonlinear dispersive media. In another limit, β = δ = 0, the equation for real Ψ, reduces to the modified Korteweg‐de Vries equation. Hence, the solutions reveal the close relation between classical solitons and envelope‐solitons.

An analysis of the perturbed plane-wave solutions of the cubic
Schroedinger equation is presented. The inverse-scattering method is
used, and the asymptotic state is represented by a series of solitons,
noting that no soliton exists whose amplitude is stationary in time. It
is suggested that the same analysis can be applied to other evolution
equations and for solving the perturbed periodic wave equation. It is
concluded that in this solution, the Jost functions should be replaced
by the Bloch eigenfunction, and many branch cuts can be made (e.g., a
cnoidal wave corresponds to two branch cuts).

We have numerically calculated chaotic waves of the focusing nonlinear Schrr̈odinger equation (NLSE), starting with a plane wave modulated by relatively weak random waves. We show that the peaks with highest amplitude of the resulting wave composition (rogue waves) can be described in terms of exact solutions of the NLSE in the form of the collision of Akhmediev breathers.

Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrödinger (NLS) equation are enumerated.
Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al .[10].
In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

We analytically investigated two-dimensional localized nonlinear waves in Kerr media with radial and azimuthal modulation of the nonlinearity and in the presence of an external potential. The solutions have been derived through the similarity transformation. We demonstrate that the properties of nonlinear waves are determined by two parameters: a whole number n (the index of the Jacobi elliptical waves) and an integer m (the topological charge). Our results indicate that the dynamic evolution, including cnoidal and snoidal waves, can be strongly affected by these two parameters, providing an approach to controlling nonlinear waves by an appropriate radial–azimuthal modulation of the nonlinearity, with an appropriate external potential.

The title (WANDT) can be applied to two objects: rogue waves in the ocean and rational solutions of the nonlinear Schrödinger equation (NLSE). There is a hierarchy of rational solutions of ‘focussing’ NLSE with increasing order and with progressively increasing amplitude. As the equation can be applied to waves in the deep ocean, the solutions can describe “rogue waves” with virtually infinite amplitude. They can appear from smooth initial conditions that are only slightly perturbed in a special way, and are given by our exact solutions. Thus, a slight perturbation on the ocean surface can dramatically increase the amplitude of the singular wave event that appears as a result.

The time evolution of a uniform wave train with a small modulation which grows is computed with a fully nonlinear irrotational flow solver. Many numerical runs have been performed varying the initial steepness of the wave train and the number of waves in the imposed modulation. It is observed that the energy becomes focussed into a short group of steep waves which either contains a wave which becomes too steep and therefore breaks or otherwise having reached a maximum modulation then recedes until an almost regular wave train is recovered. This latter case typically occurs over a few hundred time periods. We have also carried out some much longer computations, over several thousands of time periods in which several steep wave events occur. Several features of these modulations are consistent with analytic solutions for modulations using weakly nonlinear theory, which leads to the nonlinear Schrödinger equation. The steeper events are shorter in both space and time than the lower events. Solutions of the nonlinear Schrödinger equation can be transformed from one steepness to another by suitable scaling of the length and time variables. We use this scaling on the modulations and find excellent agreement particularly for waves that do not grow too steep. Hence the number of waves in the initial modulation becomes an almost redundant parameter and allows wider use of each computation. A potentially useful property of the nonlinear Schrödinger equation is that there are explicit solutions which correspond to the growth and decay of an isolated steep wave event. We have also investigated how changing the phase of the initial modulation effects the first steep wave event that occurs. c 1999 Elsevier Science B.V. All rights reserved.

Rogue waves are rare “giant”, “freak”, “monster” or “steep wave” events in nonlinear deep water gravity waves which occasionally rise up to surprising heights above the background wave field. Holes are deep troughs which occur before and/or after the largest rogue crests. The dynamical behavior of these giant waves is here addressed as solutions of the nonlinear Schrödinger equation in both 1+1 and 2+1 dimensions. We discuss analytical results for 1+1 dimensions and demonstrate numerically, for certain sets of initial conditions, the ubiquitous occurrence of rogue waves and holes in 2+1 spatial dimensions. A typical wave field evidently consists of a background of stable wave modes punctuated by the intermittent upthrusting of unstable rogue waves.

The conventional definition of rogue waves in the ocean is that their heights, from crest to trough, are more than about twice the significant wave height, which is the average wave height of the largest one-third of nearby waves. When modeling deep water waves using the nonlinear Schrödinger equation, the most likely candidate satisfying this criterion is the so-called Peregrine solution. It is localized in both space and time, thus describing a unique wave event. Until now, experiments specifically designed for observation of breather states in the evolution of deep water waves have never been made in this double limit. In the present work, we present the first experimental results with observations of the Peregrine soliton in a water wave tank.

We present a method for finding the hierarchy of rational solutions of the self-focusing nonlinear Schrödinger equation and present explicit forms for these solutions from first to fourth order. We also explain their relation to the highest amplitude part of a field that starts with a plane wave perturbed by random small amplitude radiation waves. Our work can elucidate the appearance of rogue waves in the deep ocean and can be applied to the observation of rogue light pulse waves in optical fibers.

We obtain exact spatiotemporal periodic traveling wave solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation with distributed coefficients. We utilize these solutions to construct analytical light bullet soliton solutions of nonlinear optics.

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