[Show abstract][Hide abstract] ABSTRACT: We present an infinite nonlinear Schrödinger equation hierarchy of integrable equations, together with the recurrence relations defining it. To demonstrate integrability, we present the Lax pairs for the whole hierarchy, specify its Darboux transformations and provide several examples of solutions. These resulting wavefunctions are given in exact analytical form. We then show that the Lax pair and Darboux transformation formalisms still apply in this scheme when the coefficients in the hierarchy depend on the propagation variable (e.g., time). This extension thus allows for the construction of complicated solutions within a greatly diversified domain of generalised nonlinear systems.
Chaos An Interdisciplinary Journal of Nonlinear Science 10/2015; 25(10):103114. DOI:10.1063/1.4931710 · 1.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We have found a strongly pulsating regime of dissipative solitons in the laser model described by the complex cubic-quintic Ginzburg-Landau equation. The pulse energy within each period of pulsations may change more than two orders of magnitude. The soliton spectra in this regime also experience large variations. Period doubling phenomena and chaotic behaviors are observed in the boundaries of existence of these pulsating solutions.
Physical Review E 08/2015; 92(2). DOI:10.1103/PhysRevE.92.022926 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We have found new dissipative soliton in the laser model described by the complex cubic-quintic Ginzburg-Landau equation. The soliton periodically generates spikes with extreme amplitude and short duration. At certain range of the equation parameters, these extreme spikes appear in pairs of slightly unequal amplitude. The bifurcation diagram of spike amplitude versus dispersion parameter reveals the regions of both regular and chaotic evolution of the maximal amplitudes.
[Show abstract][Hide abstract] ABSTRACT: We have found new dissipative solitons in the laser model described by the complex cubic-quintic Ginzburg- Landau equation. These objects can be called "spiny solitons" because they chaotically generate spikes of extreme amplitude and ultrashort duration.We have calculated the probability density function of these spikes that demonstrate their rogue wave nature. We have also calculated the average profiles, autocorrelation functions, average spectra, and cross-correlation frequency-resolved optical gating diagram of the "spiny solitons." They have no analogs among the noise-like pulses studied previously in experiments.
Journal of the Optical Society of America B 07/2015; 32(7):1377. DOI:10.1364/JOSAB.32.001377 · 1.97 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We analyze the quintic integrable equation of the nonlinear Schrödinger hierarchy that includes fifth-order dispersion with matching higher-order nonlinear terms. We show that a breather solution of this equation can be converted into a nonpulsating soliton solution on a background. We calculate the locus of the eigenvalues on the complex plane which convert breathers into solitons. This transformation does not have an analog in the standard nonlinear Schrödinger equation. We also study the interaction between the new type of solitons, as well as between breathers and these solitons.
Physical Review E 03/2015; 91(3-1):032928. DOI:10.1103/PhysRevE.91.032928 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We analyze the rogue wave spectra of the Sasa-Satsuma equation and their appearance in the spectra of chaotic wave fields produced through modulation instability. Chaotic wave fields occasionally produce high peaks that result in a wide triangular spectrum, which could be used for rogue wave detection.
[Show abstract][Hide abstract] ABSTRACT: We present breather solutions of the quintic integrable equation of the Schrödinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the corresponding nonlinear Schrödinger equation (NLSE) solutions. However, the presence of a free parameter in the equation results in specific solutions that have no analogues in the NLSE case. We analyze new features of these solutions.
Physical Review E 02/2015; 91(2-1):022919. DOI:10.1103/PhysRevE.91.022919 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: It seems to be self-evident that stable optical pulses cannot be considerably shorter than a single oscillation of the carrier field. From the mathematical point of view the solitary solutions of pulse propagation equations should loose stability or demonstrate some kind of singular behavior. Typically, an unphysical cusp develops at the soliton top, preventing the soliton from being too short. Consequently, the power spectrum of the limiting solution has a special behavior: the standard exponential decay is replaced by an algebraic one. We derive the shortest soliton and explicitly calculate its spectrum for the so-called short pulse equation. The latter applies to ultra-short solitons in transparent materials like fused silica that are relevant for optical fibers.
[Show abstract][Hide abstract] ABSTRACT: We report the first experimental observation of periodic breathers in water waves. One of them is Kuznetsov-Ma soliton and another one is Akhmediev breather. Each of them is a localized solution of the nonlinear Schrödinger equation (NLS) on a constant background. The difference is in localization which is either in time or in space. The experiments conducted in a water wave flume show results that are in good agreement with the NLS theory. Basic features of the breathers that include the maximal amplitudes and spectra are consistent with the theoretical predictions.
Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 10/2014; 372(2027). DOI:10.1098/rsta.2014.0005 · 2.15 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We review recent achievements in theory of ultra-short optical pulses propagating in nonlinear fibers. The following problem is especially emphasized: what is the shortest duration (the highest peak power) of an optical soliton and which physical phenomenon is responsible for breakdown of too short pulses. We argue that there is an universal mechanism that destroys sub-cycle solitons even for the most favorable dispersion profile.
[Show abstract][Hide abstract] ABSTRACT: We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.
Physical Review E 09/2014; 90(3-1):032922. DOI:10.1103/PhysRevE.90.032922 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We study the properties of the chaotic wave fields generated in the frame of the Sasa-Satsuma equation (SSE). Modulation instability results in a chaotic pattern of small-scale filaments with a free parameter-the propagation constant k. The average velocity of the filaments is approximately given by the group velocity calculated from the dispersion relation for the plane-wave solution. Remarkably, our results reveal the reason for the skewed profile of the exact SSE rogue-wave solutions, which was one of their distinctive unexplained features. We have also calculated the probability density functions for various values of the propagation constant k, showing that probability of appearance of rogue waves depends on k.
Physical Review E 09/2014; 90(3-1):032902. DOI:10.1103/PhysRevE.90.032902 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider a combined model of dissipative solitons that are generated due to the balance between gain and loss of energy as well as to the balance between input and output of matter.
[Show abstract][Hide abstract] ABSTRACT: Exploding solitons can be found in dissipative systems including laser cavities and reaction-diffusion systems. Certain choice of laser parameters allows us to obtain very high amplitudes during explosions. They can be considered as rogue waves.
[Show abstract][Hide abstract] ABSTRACT: Optical solitons cannot go beyond the critical single-cycle duration even for the most favorable medium dispersion. A universal feature that prohibits existence of too short solitons is an unavoidable formation of the singular cusp.
[Show abstract][Hide abstract] ABSTRACT: We have found experimentally and numerically, that the Fermi Pasta Ulam (FPU) recurrence is strongly influenced by the third-order dispersion (TOD) term. Namely, its presence leads to several disappearances and recoveries of the FPU recurrence when the central frequency of the pump wave is varied. The effect is highly non-trivial and can be explained in terms of reversible and irreversible losses caused by Cherenkov radiations interacting with signal harmonics of the modulation instability process.
[Show abstract][Hide abstract] ABSTRACT: We derive exact and approximate localized solutions for the Manakov-type continuous and discrete equations. We establish the correspondence between the solutions of the coupled Ablowitz-Ladik equations and the solutions of the coupled higher-order Manakov equations.
Physical Review E 07/2014; 90(1-1). DOI:10.1103/PhysRevE.90.012902 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider an extended nonlinear Schrödinger equation with higher-order odd and even terms with independent variable coefficients. We demonstrate its integrability, provide its Lax pair, and, applying the Darboux transformation, present its first and second order soliton solutions. The equation and its solutions have two free parameters. Setting one of these parameters to zero admits two limiting cases: the Hirota equation on the one hand and the Lakshmanan–Porsezian–Daniel (LPD) equation on the other hand. When both parameters are zero, the limit is the nonlinear Schrödinger equation.
Physics Letters A 01/2014; 378(4):358–361. DOI:10.1016/j.physleta.2013.11.031 · 1.68 Impact Factor