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On the Nonlinear Schrödinger Equation on the Half Line with Homogeneous Robin Boundary Conditions
Abstract
Boundary value problems for the nonlinear Schrödinger equations on the half line with homogeneous Robin boundary conditions are revisited using Bäcklund transformations. In particular: relations are obtained among the norming constants associated with symmetric eigenvalues; a linearizing transformation is derived for the Bäcklund transformation; the reflection‐induced soliton position shift is evaluated and the solution behavior is discussed. The results are illustrated by discussing several exact soliton solutions, which describe the soliton reflection at the boundary with or without the presence of self‐symmetric eigenvalues.
... with α an arbitrary real constant. Importantly, it was also shown in [11,12,13,14] that, when such linearizable BCs are given, the BVP acquires extra symmetries. In particular, using a nonlinear version of the method of images, it was shown in [13,14] that the discrete eigenvalues of the associated scattering problem appear in symmetric quartets, and that to each soliton in the physical domain 0 < X < ∞ there corresponds a mirror soliton in the virtual (reflected) domain −∞ < X < 0. This nonlinear method of images was then later extended and applied to other discrete and continuous NLS-type systems in [15,16,17]. ...
... Importantly, it was also shown in [11,12,13,14] that, when such linearizable BCs are given, the BVP acquires extra symmetries. In particular, using a nonlinear version of the method of images, it was shown in [13,14] that the discrete eigenvalues of the associated scattering problem appear in symmetric quartets, and that to each soliton in the physical domain 0 < X < ∞ there corresponds a mirror soliton in the virtual (reflected) domain −∞ < X < 0. This nonlinear method of images was then later extended and applied to other discrete and continuous NLS-type systems in [15,16,17]. ...
... The product in the right-hand side of Eq. (2.24) is nothing but the value of 1/s 11 (k j ) in the reflectionless case, as obtained from the trace formulae, and will make some key formulae for the BVP much simpler than the corresponding ones in [13,14] (cf. section 5). ...
Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coordinates are considered. A class of boundary conditions that lead to linearizable problems is identified by introducing appropriate extensions to initial-value problems on the infinite line, either explicitly or by constructing a suitable Bäcklund transformation. Various soliton solutions are explicitly constructed and studied.
... with α an arbitrary real constant. Importantly, it was also shown in [11,12,13,14] that, when such linearizable BCs are given, the BVP acquires extra symmetries. In particular, using a nonlinear version of the method of images, it was shown in [13,14] that the discrete eigenvalues of the associated scattering problem appear in symmetric quartets, and that to each soliton in the physical domain 0 < X < ∞ there corresponds a mirror soliton in the virtual (reflected) domain −∞ < X < 0. This nonlinear method of images was then later extended and applied to other discrete and continuous NLS-type systems in [15,16,17]. ...
... Importantly, it was also shown in [11,12,13,14] that, when such linearizable BCs are given, the BVP acquires extra symmetries. In particular, using a nonlinear version of the method of images, it was shown in [13,14] that the discrete eigenvalues of the associated scattering problem appear in symmetric quartets, and that to each soliton in the physical domain 0 < X < ∞ there corresponds a mirror soliton in the virtual (reflected) domain −∞ < X < 0. This nonlinear method of images was then later extended and applied to other discrete and continuous NLS-type systems in [15,16,17]. ...
... The product in the right-hand side of Eq. (2.24) is nothing but the value of 1/s 11 (k j ) in the reflectionless case, as obtained from the trace formulae, and will make some key formulae for the BVP much simpler than the corresponding ones in [13,14] (cf. section 5). ...
Boundary value problems for the nonlinear Schrodinger equation on the half line in laboratory coordinates are considered. A class of boundary conditions that lead to linearizable problems is identified by introducing appropriate extensions to initial-value problems on the infinite line, either explicitly or by constructing a suitable Backlund transformation. Various soliton solutions are explicitly constructed and studied.
... This approach obviously differs from the standard IST in which the spectral analysis of only one part of the Lax pairs was considered [9]. The Fokas' unified method has been used to explore boundary value problems of some physically significant integrable nonlinear evolution equations (NLEEs) with 2 × 2 Lax pairs on the half-line and the finite interval (e.g., the nonlinear Schrödinger equation [6,[11][12][13][14], the sine-Gordon equation [15,16], the KdV equation [17], the mKdV equation [18][19][20], the derivative nonlinear Schrödinger equation [21,22], Ernst equations [23,24], and etc. [25][26][27][28][29][30][31]) and ones with 3 × 3 Lax pairs on the half-line and the finite interval (e.g., [32], the Degasperis-Procesi equation [33], the Sasa-Satsuma equation [34], the coupled nonlinear Schrödinger equations [35][36][37][38], and the Ostrovsky-Vakhnenko equation [39]). ...
... Notice that the symmetric matrix P used here differs from the diag ones used in 3 × 3 Lax pairs [35][36][37][38]. Similar to the proof in Ref. [14], based on Eq. (24) and (27) we have the following proposition: ...
... which leads to S(k) and s T (k) in terms of Eqs. (35) and (36), where µ j2 (0, t, k), j 2 = 1, 2, µ j3 (L, t, k), j 3 = 3, 4, µ j1 (x, 0, k), j 1 = 2, 3, 4, µ 2 (x, T, k), 0 < x < L, 0 < t < T are defined by the integral equations ...
We investigate the initial-boundary value problem for the general three-component nonlinear Schrodinger (gtc-NLS) equation with a 4x4 Lax pair on a finite interval by extending the Fokas unified approach. The solutions of the gtc-NLS equation can be expressed in terms of the solutions of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Moreover, the relevant jump matrices of the RH problem can be explicitly found via the three spectral functions arising from the initial data, the Dirichlet-Neumann boundary data. The global relation is also established to deduce two distinct but equivalent types of representations (i.e., one by using the large k of asymptotics of the eigenfunctions and another one in terms of the Gelfand-Levitan-Marchenko (GLM) method) for the Dirichlet and Neumann boundary value problems. Moreover, the relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval approaches to infinity. Finally, we also give the linearizable boundary conditions for the GLM representation.
... The numerical experiments reported in Section 3 are good examples where such a space-time adaptive algorithm will be very beneficial. Due to the integrability properties of the CNLS equation, the reflection of a soliton can be studied analytically only in 1D, [4,5,14]. In this paper, after verifying the order of accuracy of the numerical method in space and time, we validate the efficiency of the numerical method by studying first the perfect (elastic) reflection of dark and bright solitons on vertical walls using either (3) or (4) as boundary conditions [4]. ...
... The reflection is called perfect if the reflected wave has the same shape as the original soliton but different direction of propagation. This behaviour has been studied analytically in [4,5,14] for the integrable CNLS equation in one space dimension. We performed two numerical tests using boundary conditions (3) and (4). ...
... Both type of boundary conditions will give perfect reflections, but the interaction of the soliton with the boundary is different. For the differences between the two reflections, we refer to [4,5]. & Sons, Ltd. ...
... Moreover, the relaxation scheme exhibits mass conservation, same like the standard Crank-Nicolson scheme, thus preserving the mass conservation property of the continuous problem, cf. (5). However, the relaxation scheme does not preserve the energy E(·) in two space dimensions. ...
... Due to the integrability properties of the CNLS equation, the reflection of a soliton can be studied analytically only in 1D, [4,5,14]. In this paper after verifying the order of accuracy of the numerical method in space and time, we validate the efficiency of the numerical method by studying first the perfect (elastic) reflection of dark and bright solitons on vertical walls using either (3) or (4) as boundary conditions, [4]. ...
... The reflection is called perfect if the reflected wave has the same shape as the original soliton but different direction of propagation. This behaviour has been studied analytically in [4,5,14] for the integrable CNLS equation in one space dimension. We performed two numerical tests using boundary conditions (3) and (4). ...
In this paper we perform a numerical study on the interesting phenomenon of soliton reflection of solid walls. We consider the 2D nonlinear Schrodinger equation as the underlying mathematical model and we use an implicit-explicit type Crank-Nicolson finite element scheme for its numerical solution. After verifying the perfect reflection of the solitons on a vertical wall, we present the imperfect reflection of a dark soliton on a diagonal wall.
... Limiting cases are Neumann and Dirichlet BCs, obtained for a = 0 and as a ¥ in (1.2), respectively. In [10] and [7], the BVP with linearizable BCs was considered using the extensions of the potential to the whole real line introduced in [14] and in [17,24], respectively, and it was shown, that the discrete eigenvalues of the scattering problem appear in symmetric quartets-as opposed to pairs in the initial value problem (IVP). Moreover, the symmetries of the discrete spectrum, norming constants, reflection coefficients and scattering data were obtained, and the reflection experienced by the solitons at the boundary was explained as a special form of soliton interaction. ...
... In this section we obtain a discrete analogue of the BT that was used in [6,7,12] to solve the BVP for the NLS equation with linearizable BCs. We begin with the following result, the proof of which is obtained by direct calculation: )such that the corresponding eigenfunctions F n t z , , n ( ) and F n t z , , ñ ( ) satisfy F = F n t z B n t z n t z , , , ...
... Importantly, (7.10) implies that an additional symmetry exists for the discrete spectrum in the BVP compared to the IVP, similarly to in the continuum case. Recalling that the discrete eigenvalues are the zeros of a z 11 ( ) and a z 22 ( ), we have: In the first case (i.e., real eigenvalues), these eigenvalues correspond to the self-symmetric eigenvalues of the continuum problem [7], and we will refer to them as self-symmetric eigenvalues of the AL system. In the second case (i.e., purely imaginary eigenvalues), these eigenvalues arise as an artifact of the discretization, and have no correspondence in the continuum case. ...
The boundary value problem (BVP) for the Ablowitz-Ladik (AL) system on the natural numbers with linearizable boundary conditions is studied. In particular: (i) a discrete analogue is derived of the Bcklund transformation that was used to solved similar BVPs for the nonlinear Schrödinger equation; (ii) an explicit proof is given that the Bcklund-transformed solution of AL remains within the class of solutions that can be studied by the inverse scattering transform; (iii) an explicit linearizing transformation for the Bcklund transformation is provided; (iv) explicit relations are obtained among the norming constants associated with symmetric eigenvalues; (v) conditions for the existence of self-symmetric eigenvalues are obtained. The results are illustrated by several exact soliton solutions, which describe the soliton reflection at the boundary with or without the presence of self-symmetric eigenvalues.
... Recently, Lenells extended the Fokas method to study the initial-boundary value (IBV) problems for integrable nonlinear evolution equations with 3 × 3 Lax pairs on the half-line [22]. After that, the idea was extended to study IBV problems of some integrable nonlinear evolution equations with 3 × 3 Lax pairs on the half-line or the finite interval, such as the Degasperis-Procesi equation [23], the Sasa-Satsuma equation [24], the coupled nonlinear Schrödinger equations [25][26][27][28], and the Ostrovsky-Vakhnenko equation [29]. To the best of our knowledge, so far there is no work on the IBV problems of integrable equations with 4 × 4 Lax pairs on the half-line. ...
... Thus, it follows from Eqs. (25) and (26) that s(k) and S(k) are determined by U (x, 0, k) and V (0, t, k), i.e., by the initial data q j (x, t = 0) and the Dirichlet-Neumann boundary data q j (x = 0, t) and q jx (x = 0, t), j = 1, 0, −1, respectively. In fact, µ 3 (x, 0, k) and µ 1,2 (0, t, k) satisfy the x-part and t-part of the Lax pair (7) at t = 0 and x = 0, respectively, that is, ...
We investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii (GP) equations with a 4x4 Lax pair on the half-line. The solution of this system can be obtained in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x=0. The global relation is established between the two dependent spectral functions. The general mappings between Dirichlet and Neumann boundary values are analyzed in terms of the global relation.
... In [8], the authors showed that there exists a unique global classical solution of the forced NLS equation (1) in C 0 (R + ; H 2 (R + )) ∩ C 1 (R + ; L 2 (R + )), when φ ∈ H 2 (R + ) and h ∈ C 2 (R + ). In the paper [2], the boundary value problem for the nonlinear Schr"odinger equation on the half-line with homogeneous Robin boundary conditions was studied via Backlund transformations. Bona, Sun, and Zang, in [4], obtained local well-posedness when the initial data lie in Sobolev spaces H s (R + ) for s > 1 2 ; also, global well-posedness is established for s ≥ 1. ...
... (0, T ). Moreover V. Kalantarov and T.Özsar, in [29], study the interaction between a nonlinear focusing Robin type boundary source, a nonlinear defocusing interior source, and a weak damping term for nonlinear Schrödinger equations (2). They construct solutions with negative initial energy satisfying a certain set of conditions which blow-up in finite time in the H 1sense. ...
In this paper, we study the initial-boundary-value problem for the cubic nonlinear Schrödinger equation, formulated on a half-line with different inhomogeneous boundaries data. We study the well-posedness in the case of Neumann and Robin condition in Sobolev space of low regularity. Also, we revisit, in a self-consistent way, some results concerning the Dirichlet condition.
... The effectiveness of the method was promptly illustrated on the nonlinear Schrödinger equation (NLS) in [8]. This led to a wealth of subsequent activity, see for instance [35,11,9,22,18,15] for NLS, and became known as nonlinear mirror image method following the terminology of [11]. Let us note that prior to this important development, the idea of the mirror image method had been used in [1] to solve NLS with Dirichlet or Neumann BCs only, by a direct application of the odd/even extension method. ...
... • The zeros of a(λ) are composed of p pairs (λ k , −λ * k ), k = 1, . . . , p and s self-symmetric zeros [9] λ k = iσ k ∈ iR + , k = 1, . . . , s. ...
We perform the analysis of the focusing nonlinear Schrödinger equation on the half-line with time-dependent boundary conditions along the lines of the nonlinear method of images with the help of Bäcklund transformations . The difficulty arising from having such time-dependent boundary conditions at is overcome by changing the viewpoint of the method and fixing the Bäcklund transformation at infinity as well as relating its value at to a time-dependent reflection matrix. The interplay between the various aspects of integrable boundary conditions is reviewed in detail to brush a picture of the area. We find two possible classes of solutions. One is very similar to the case of Robin boundary conditions whereby solitons are reflected at the boundary, as a result of an effective interaction with their images on the other half-line . The new regime of solutions supports the existence of one soliton that is not reflected at the boundary but can be either absorbed or emitted by it. We demonstrate that this is a unique feature of time-dependent integrable boundary conditions.
... Theorem I. 4. Let α, ω ∈ R be such that ω > α 2 . ...
... This solution is the unique positive minimizer of (2.11). Furthermore, we have an explicit formula forφ ω ϕ ω (x) = 2 4 √ ω sech ...
We consider the Schrödinger equation with a nonlinear derivative term on [0, +∞) under the Robin boundary condition at 0. Using a virial argument, we obtain the existence of blowing up solutions, and using variational techniques, we obtain stability and instability by blow-up results for standing waves.
... However, in (2 + 1)-dimensions, these systems lose their integrability and hence one depends on numerical techniques for exploring the phenomena of solitons reflections and collisions. In literature, there are few researches that analytically investigate the reflection and collisions of solitons for (1 + 1)dimensional systems because of the integrability properties [10][11][12][13][14]. Various numerical approaches have been presented for solving nonlinear Schrödinger-type equations [15][16][17][18][19][20]. ...
... where λ, γ are real constants, and ξ is the amplification vector. Substituting of (12) into system (11) leads to the equation ...
In this work, we develop an efficient numerical scheme based on the method of lines (MOL) to investigate the interesting phenomenon of collisions and reflections of optical solitons. The established scheme is of second order in space and of fourth order in time with an explicit nature. We deduce stability restrictions using the von Neumann stability analysis. We consider a ( 2 + 1 ) -dimensional system of a coupled nonlinear Schrödinger equation as a general model of nonlinear Schrödinger-type equations. We consider several numerical experiments to demonstrate the robustness of the scheme in capturing many scenarios of collisions and reflections of the optical solitons related to nonlinear Schrödinger-type equations. We verify the scheme accuracy through computing the conserved invariants and comparing the present results with some existing ones in the literature.
... The method does not artificially truncate infinite physical domains. 6. The solution steps require only the solution of linear problems. ...
... Meanwhile, the long time behavior of such solutions at x = 0 is dominated by oscillatory standing solitons leading to non-decaying boundary data. The choice of the sign of ρ is also related to the possibility of extending the half-line solution to a bounded whole-line solution, see [6,18] for further details. ...
We implement the Numerical Unified Transform Method to solve the Nonlinear Schr\"odinger equation on the half-line. For so-called linearizable boundary conditions, the method solves the half-line problems with comparable complexity as the Numerical Inverse Scattering Transform solves whole-line problems. In particular, the method computes the solution at any x and t without spatial discretization or time stepping. Contour deformations based on the method of nonlinear steepest descent are used so that the method's computational cost does not increase for large x,t and the method is more accurate as x,t increase. Our ideas also apply to some cases where the boundary conditions are not linearizable.
... The effectiveness of the method was promptly illustrated on the nonlinear Schrödinger equation (NLS) in [8]. This led to a wealth of subsequent activity, see for instance [35,11,9,22,18,15] for NLS, and became known as nonlinear mirror image method following the terminology of [11]. Let us note that prior to this important development, the idea of the mirror image method had been used in [1] to solve NLS with Dirichlet or Neumann BCs only, by a direct application of the odd/even extension method. ...
... • The zeros of a(λ) are composed of p pairs (λ k , −λ * k ), k = 1, . . . , p and s self-symmetric zeros [9] λ k = iσ k ∈ iR + , k = 1, . . . , s. ...
We perform the analysis of the focusing nonlinear Schr\"odinger equation on the half-line with time-dependent boundary conditions along the lines of the nonlinear method of images with the help of B\"acklund transformations. The difficulty arising from having such time-dependent boundary conditions at x=0 is overcome by changing the viewpoint of the method and fixing the B\"acklund transformation at infinity as well as relating its value at x=0 to a time-dependent reflection matrix. The interplay between the various aspects of integrable boundary conditions is reviewed in detail to brush a picture of the area. We find two possible classes of solutions. One is very similar to the case of Robin boundary conditions whereby solitons are reflected at the boundary, as a result of an effective interaction with their images on the other half-line. The new regime of solutions supports the existence of one soliton that is not reflected at the boundary but can be either absorbed or emitted by it. We demonstrate that this is a unique feature of time-dependent integrable boundary conditions.
... This reflects the space inverse symmetry of NLS and allows to obtain solutions of the model using the usual IST by uniquely looking at the positive semi axis. The technique was applied to the vector NLS model [24], and was used to obtain boundarybound solitons [25] that are static solitons subject to the BCs. Note that a recent study of the model was reported in [26] following a functional analysis approach. ...
... Sinceλ j = −λ j , each ψ j needs to be paired with itself to eventually preserve to the boundary constraint (4.3) in dressing the boundary. Similar results were obtained in [25] using the mirror-image technique. Proposition 5.1 (Boundary-bound solitons) Assume that, associated with N distinct pure imaginary parameters λ j = iκ j , κ j ∈ R, there exist N special solutions ψ j (λ j ), j = 1, . . . ...
Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schr\"odinger (NLS) equation on the half-line. The integrable BCs at the origin are represented by constraints of the Lax pair, and our method lies on dressing the Lax pair by preserving those constraints in the Darboux-dressing process. The method is applied to two classes of solutions: solitons vanishing at infinity and self-modulated solitons on a constant background. Half-line solitons in both cases are explicitly computed. In particular, the boundary-bound solitons, that are static solitons bounded at the origin, are also constructed. We give a natural inverse scattering transform interpretation of the method as the scattering data determined by the integrable BCs evolve linearly in space.
... Recently, Lenells extended the Fokas method to study the initial-boundary value (IBV) problems for integrable nonlinear evolution equations with 3 × 3 Lax pairs on the half-line [22]. After that, the idea was extended to study IBV problems of some integrable nonlinear evolution equations with 3 × 3 Lax pairs on the half-line or the finite interval, such as the Degasperis-Procesi equation [23], the Sasa-Satsuma equation [24], the coupled nonlinear Schrödinger equations [25][26][27][28], and the Ostrovsky-Vakhnenko equation [29]. To the best of our knowledge, so far there is no work on the IBV problems of integrable equations with 4 × 4 Lax pairs on the half-line. ...
... Thus, it follows from Eqs. (25) and (26) that s(k) and S(k) are determined by U (x, 0, k) and V (0, t, k), i.e., by the initial data q j (x, t = 0) and the Dirichlet-Neumann boundary data q j (x = 0, t) and q jx (x = 0, t), j = 1, 0, −1, respectively. In fact, µ 3 (x, 0, k) and µ 1,2 (0, t, k) satisfy the x-part and t-part of the Lax pair (7) at t = 0 and x = 0, respectively, that is, ...
We investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii (GP) equations with a 4x4 Lax pair on the half-line. The solution of this system can be obtained in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x=0. The global relation is established between the two dependent spectral functions. The general mappings between Dirichlet and Neumann boundary values are analyzed in terms of the global relation.
... It was proved that the solution has different asymptotic behaviors in different regions. In paper [1] boundary value problems for the nonlinear Schrödinger equations on the half-line with homogeneous Robin boundary conditions were revisited using Bäcklund transformations. The results were illustrated by discussing several exact soliton solutions, which described the soliton reflection at the boundary. ...
... 1) where N (u) = |u| 2 u. We use the notation Z := H 1 (R + ) ∩ H 0,1 1 (R + ), Y = H Z := φ H 1 + φ H 0,1Let us define the functional spaceX := φ ∈ C [0, ∞); H 1 (R + ) ∩ H 0,1 (R + ) : φ X < ∞ , (t) H 1 + φ(t) H 0,1 + t ρ φ(t) L ∞ ,where ρ = min{2, β} > 1 2 . ...
We consider the initial–boundary value problem for the fractional Schrödinger equation, posed on positive half-line x>0:{ut+iuxx+i|u|2u+|∂x|12u=0,t≥0,x≥0;u(x,0)=u0(x),x>0,ux(0,t)=h(t),t>0,
where |∂x|12 is the fractional derivative operator defined by the Riesz potential|∂x|12=12π∫0∞sign(x−y)|x−y|uy(y)dy.
We study the global existence in time and asymptotics of solutions to the initial–boundary value problem.
... Also note that, since the potential (4.1) is even, the IVP is equivalent to an initialboundary value problem on the half-line with homogeneous Neumann boundary conditions. For decaying potentials at infinity, such problems were studied in [5,10,11,9,22,32,45]. In Appendix B we discuss the extension of those results to the case of potentials with NZBC at infinity. ...
... In the case of an even potential (i.e.,Q(x, t) = Q(x, t)) we haveφ ± (x, t, k) = φ ± (x, t, k) andS(k) = S(k), so s 1,1 (k) = s 2,2 (−k) = s * 1,1 (−k * ). Thus, discrete eigenvalues appear in symmetric quartets, i.e., ±k n , ±k * n , or, when formulated in the uniformization variable, in symmetric octets, as in the boundary value problem with ZBC [10,11]. This allows us to confine our search for discrete eigenvalues to C I in Theorems 5.1 and 5.4. ...
We investigate the nonlinear stage of the modulational (or Benjamin-Feir) instability by characterizing the initial value problem for the focusing nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBC) at infinity. We do so using the recently formulated inverse scattering transform (IST) for this problem. While the linearization of the NLS equation ceases to be valid when the perturbations have grown sufficiently large compared to the background, the results of the IST remain valid for all times and therefore provide a convenient way to study the nonlinear stage of the modulational instability. We begin by studying the spectral problem for the Dirac operator (i.e., the first half of the Lax pair for the NLS equation) with piecewise constant initial conditions which are a generalization to NZBC of a potential well and a potential barrier. Since the scattering data uniquely determine the time evolution of the initial condition via the inverse problem, the study of these kinds of potentials provides a simple means of investigating the growth of small perturbations of a constant background via IST. We obtain several results. First, we prove that there are arbitrarily small perturbations of the constant background for which there are discrete eigenvalues, which shows that no area theorem is possible for the NLS equation with NZBC. Second, we prove that there is a class of perturbations for which no discrete eigenvalues are present. In particular, this latter result shows that solitons cannot be the primary vehicle for the manifestation of the instability, contrary to a recent conjecture. We supplement these results with a numerical study about the existence, number, and location of discrete eigenvalues in other situations. Finally, we compute the small-deviation limit of the IST, and we compare it with the direct linearization of the NLS equation around a constant background, which allows us to precisely identify the nonlinear analogue of the unstable Fourier modes within the IST. These are the Jost eigenfunctions for values of the scattering parameter belonging to a finite interval of the imaginary axis around the origin. Importantly, this last result shows that the IST contains an automatic mechanism for the saturation of the modulational instability.
... A method of images approach can be used to solve this problem. The approach of Biondini & Bui [27], first introduced by Bikbaev & Tarasov [28], takes the given initial condition on [0, ∞) and produces an extension to (−∞, 0) using a Darboux transformation. For Neumann boundary conditions, this results in an even extension and for Dirichlet boundary conditions the transformation produces an odd extension. ...
... It was shown in [27] that the solution of the Cauchy problem for NLS equation on R with initial dataq, restricted to [0, ∞), is the unique solution of (5.1). To compute the extended initial dataq, we first solve the system (5.2) numerically using a combination of Runge-Kutta 4 and 5. ...
We solve the focusing and defocusing nonlinear Schrödinger (NLS) equations numerically by implementing the inverse scatterin transform. The computation of the scattering data and of the NLS solution are both spectrally convergent. Initial condition in a suitable space are treated. Using the approach of Biondini & Bui, we numerically solve homogeneous Robin boundary-valu problems on the half line. Finally, using recent theoretical developments in the numerical approximation of Riemann–Hilber problems, we prove that, under mild assumptions, our method of approximating solutions to the NLS equations is uniformly accurat in their domain of definition.
... The integrability of the resulting boundary systems is ensured by the existence of infinitely many integrals of the motion in involution. Several solution methods to the integrable boundary value problems on the half-line (or the interval) have been developed, such as the so-called nonlinear mirror image method [2][3][4][5][6][7][8]16], and a boundary dressing technique presented recently in [9][10][11]. ...
We study integrable boundary conditions associated with the whole hierarchy of nonlinear Schr\"{o}dinger (NLS) equations defined on the half-line. We find that the even order NLS equations and the odd order NLS equations admit rather different integrable boundary conditions. In particular, the odd order NLS equations permit a new class of integrable boundary conditions that involves the time reversal. We prove the integrability of the NLS hierarchy in the presence of our new boundary conditions in the sense that the models possess infinitely many integrals of the motion in involution. Moreover, we develop further the boundary dressing technique to construct soliton solutions for our new boundary value problems.
... Recently, Lenells further developed the Fokas method to analyze the IBV problems for integrable NLEEs with 3 × 3 Lax pairs on the half-line [32]. After that, the modified approach was applied in the IBV problems of other integrable NLEEs with 3 × 3 Lax pairs on the half-line or the finite interval, such as the Degasperis-Procesi equation [33], the Sasa-Satsuma equation [34], the coupled NLS equations [35][36][37], and the Ostrovsky-Vakhnenko equation [38]. ...
In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on the finite interval by extending the Fokas unified transform approach. The solution of this system can be expressed in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Furthermore, the relevant jump matrices with explicit (x, t)-dependence of the matrix RH problem can be explicitly found via three spectral functions {s(k), S(k), S_L(k)} arising from the initial data and the Dirichlet-Neumann boundary conditions at x=0 and x=L, respectively. The global relation is also found to deduce two distinct but equivalent types of representations (i.e., one via the large k of asymptotics of the eigenfunctions and another one in terms of the Gel'fand-Levitan-Marchenko (GLM) approach) for the Dirichlet and Neumann boundary value problems. In particular, the formulae for IBV problems on the finite interval can reduce to ones on a half-line as the length L of the interval approaches to infinity. Moreover, we also present the linearizable boundary conditions for the GLM representations.
... Refs. [41][42][43][44][45][46][47][48]. This topic is left for further study. ...
The method of nonlinearization of the Lax pair is developed for the Ablowitz-Kaup-Newell-Segur (AKNS) equation in the presence of space-inverse reductions. As a result, we obtain a new type of finite-dimensional Hamiltonian systems: they are nonlocal in the sense that the inverse of the space variable is involved. For such nonlocal Hamiltonian systems, we show that they preserve the Liouville integrability and they can be linearized on the Jacobi variety. We also show how to construct the algebro-geometric solutions to the AKNS equation with space-inverse reductions by virtue of our nonlocal finite-dimensional Hamiltonian systems. As an application, algebro-geometric solutions to the AKNS equation with the Dirichlet and with the Neumann boundary conditions, and algebro-geometric solutions to the nonlocal nonlinear Schrödinger (NLS) equation are obtained. nonlocal finite-dimensional integrable Hamiltonian system, algebro-geometric solution, Dirichlet (Neumann) boundary, nonlocal NLS equation.
... In addition to the integrability aspect, another important topic is the solution method to the integrable boundary problem. In this respect, several methods have been introduced to solve the integrable boundary value problems defined on the half-line or the finite interval, such as the nonlinear mirror image method developed e.g. in [3][4][5][6][7] following the idea initiated in [8][9][10][11], a boundary dressing technique presented recently in [12], and a more recent method of [13,14] which is based on Sklyanin's double-row monodromy matrix. The first and third methods in the aforementioned ones extend the applications of the well-known inverse scattering transform (IST) (see e.g. ...
We consider the nonlinear Schr\"{o}dinger (NLS) equation on the half-line subjecting to a class of boundary conditions preserve the integrability of the model. For such a half-line problem, the Poisson brackets of the corresponding scattering data are computed, and the variables of action-angle type are constructed. These action-angle variables completely trivialize the dynamics of the NLS equation on the half-line.
... Meanwhile, the longtime behaviour of such solutions at x = 0 is dominated by oscillatory standing solitons leading to non-decaying boundary data. The choice of the sign of ρ is also related to the possibility of extending the half-line solution to a bounded whole-line solution; see [32,33] for further details. ...
We implement the numerical unified transform method to solve the nonlinear Schrödinger equation on the half-line. For the so-called linearizable boundary conditions, the method solves the half-line problems with comparable complexity as the numerical inverse scattering transform solves whole-line problems. In particular, the method computes the solution at any x and t without spatial discretization or time stepping. Contour deformations based on the method of nonlinear steepest descent are used so that the method’s computational cost does not increase for large x , t and the method is more accurate as x , t increase. Our ideas also apply to some cases where the boundary conditions are not linearizable.
... The extension reflects the space inversion symmetry of NLS [32,41] and allows us to solve the model using the usual inverse scattering transform by uniquely looking at the positive semi axis. Generalization of this method can be found in [10,16,17]. Another direct approach, known as dressing the boundary, was introduced by the author to derive soliton solutions of NLS on the half-line [47]. ...
In this paper, we develop an inverse scattering transform for the integrable focusing nonlinear Schr\"odinger (NLS) equation on the half-line subject to a class of boundary conditions. The method is based on the notions of integrable boundary conditions and double-row monodromy matrix, developed by Sklyanin, which characterize initial-boundary value problems for NLS on an interval. It follows from Sklyanin's approach that a hierarchy of integrable boundary conditions can be encoded into a hierarchy of reflection matrices, which together with the time-part Lax matrix of NLS, form a semi-discrete type Lax pair. The inverse scattering transform relies on the formulation of a scattering system for the double-row monodromy matrix characterizing an interval problem. The scattering system for the half-line problem for NLS is obtained by extending one endpoint of the interval to infinity. Then, we derive spectral and analytic properties of the scattering systems, and set up the inverse part using a Riemann-Hilbert formulation. We also show that our approach is equivalent to a nonlinear method of reflection by extending the initial-boundary value problem on the half-line to an initial value on the whole axis. Explicit examples of soliton solutions on the half-line are provided. Although we only consider the NLS model as a particular example, the inverse scattering transform we present in this paper can be readily generalized to a wide range of integrable PDEs on the half-line.
... Our method for the construction of solutions is based on the Darboux transformation method in conjunction with a boundary dressing technique. It is worth mentioning that an analogous boundary problem (boundary involving a time derivative of the field) for the integrable discrete NLS equation was studied very recently in [11], where the nonlinear mirror image method [17,18] was applied to construct the solutions. We believe that the nonlinear mirror image method can also be applied to solve the boundary problem (1.2) for the NLS equation. ...
We study the nonlinear Schrödinger equation on the half-line with a new boundary condition presented by Zambon. This new boundary involves a time derivative of the field and was already shown by Zambon to be integrable. In this paper we re-establish the integrability of such a boundary both by using the Sklyanin’s formalism and by using the tool of Bäcklund transformations. Moreover, we present a method to derive explicit formulae for multi-soliton solutions of the boundary problem by virtue of the Darboux transformation method in conjunction with a boundary dressing technique.
... Our method for the construction of solutions is based on the Darboux transformation method in conjunction with a boundary dressing technique. It is worth mentioning that an analogous time-dependent boundary for the integrable discrete NLS equation was studied very recently in [11], where the nonlinear mirror image method [17,18] was applied to construct the solutions. We believe that the nonlinear mirror image method can also be applied to solve the boundary problem (1.2) for the NLS equation. ...
We study the nonlinear Schr\"odinger equation on the half-line with a boundary condition that involves time derivative. This boundary condition was presented by Zambon [J. High Energ. Phys. 2014 (2014) 36]. We establish the integrability of such a boundary both by using the Sklyanin's formalism and by using the tool of B\"acklund transformations together with a suitable reduction of reflection type. Moreover, we present a method to derive explicit formulae for multi-soliton solutions of the boundary problem by virtue of the Darboux transformation method in conjunction with a boundary dressing technique.
... The Cauchy problem for the cubic nonlinear Schrödinger equations was studied by many authors extensively, see [6] and references cited therein. On the other hand, there are some results on the initial boundary value problem for nonlinear Schrödinger equations with homogeneous boundary conditions (see [2], [23], [24], [9], [10], [11], [12], [26]).In paper [1] boundary value problems for the nonlinear Schrödinger equations on the half-line with homogeneous Robin boundary conditions were revisited using Bäcklund transformations. There are also some existence results on the inhomogeneous boundary value problem, see [3], [5], in one dimension and [25] in existence of weak solutions in general space dimension without uniqueness of solutions. ...
We consider the inhomogeneous Mixed-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques
... In computing the boundary-bound solitons, the expressions for the norming constants are different for the odd and even soliton numbers. One also excludes the situation where the stationary solitons are subject to the Dirichlet BCs by assuming f α (λ j ) < 0. Note that for the scalar NLS case, the boundary-bound states were investigated in [13,27]. One can put the stationary and moving solitons together by combining the associated soliton data. ...
We construct multi-soliton solutions of the n-component vector nonlinear Schr\"odinger equation on the half-line subject to two classes of integrable boundary conditions (BCs): the homogeneous Robin BCs and the mixed Neumann/Dirichlet BCs. The construction is based on the approach of dressing the integrable BCs: soliton solutions are generated in preserving the integrable BCs at each step of the Darboux-dressing process. Under the Robin BCs, examples, including boundary-bound solitons, are explicitly derived; under the mixed Neumann/Dirichlet BCs, the boundary can act as a polarizer that tunes different components of the vector solitons.
... After that it is also interesting to study the influence of the boundary data on the qualitative properties of the solution. In paper [1] boundary value problems for the nonlinear Schrödinger equations (NLS) (a 3 = 0, a 2 ∈ R) on the half-line with homogeneous Robin boundary conditions were revisited using Bäcklund transformations. These results were illustrated by discussing several exact soliton solutions, which described the soliton reflection at the boundary. ...
We study the initial-boundary value problem for the Schrödinger equation with potential, posed on positive half-line where , is the fractional derivative operator defined by the modified Risz potential We are interested in the initial-boundary value problem with small initial data and Robin boundary data given in a suitable weighted Sobolev spaces. We show that problem admits global solutions whose long-time behavior essentially depends on the boundary data.
... Recently, Lenells further developed the Fokas method to analyze the IBV problems for integrable NLEEs with 3 × 3 Lax pairs on the half-line [32]. After that, the modified approach was applied in the IBV problems of other integrable NLEEs with 3 × 3 Lax pairs on the half-line or the finite interval, such as the Degasperis-Procesi equation [33], the Sasa-Satsuma equation [34], the coupled NLS equations [35][36][37], and the Ostrovsky-Vakhnenko equation [38]. ...
In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on the finite interval by extending the Fokas unified transform approach. The solution of this system can be expressed in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Furthermore, the relevant jump matrices with explicit (x, t)-dependence of the matrix RH problem can be explicitly found via three spectral functions {s(k), S(k), S_L(k)} arising from the initial data and the Dirichlet-Neumann boundary conditions at x=0 and x=L, respectively. The global relation is also found to deduce two distinct but equivalent types of representations (i.e., one via the large k of asymptotics of the eigenfunctions and another one in terms of the Gel'fand-Levitan-Marchenko (GLM) approach) for the Dirichlet and Neumann boundary value problems. In particular, the formulae for IBV problems on the finite interval can reduce to ones on a half-line as the length L of the interval approaches to infinity. Moreover, we also present the linearizable boundary conditions for the GLM representations.
... In 2012, Lenells [26] implemented the Fokas method to IBV problems for integrable evolution equations with Lax pair equations involving 3 × 3 matrices on the half-line. This method is also used to analyse the Degasperis-Procesi [27], Sasa-Satsuma [28], three-wave [29] and the nonlinear Schrödinger equations [30][31][32][33][34]. ...
In this paper, we implement the Fokas method to study initial-boundary value problems of the mixed coupled nonlinear Schrödinger equation formulated on the half-line with Lax pairs involving 3 × 3 matrices. The solution can be written in terms of the solution to a 3 × 3 Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions s(k) and S(k), which are determined by the initial values and boundary values at x=0, respectively. Some of these boundary values are unknown; however, using the fact that these specific functions satisfy a certain global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data. © 2016 The Author(s) Published by the Royal Society. All rights reserved.
... It was proved that the solution exhibits different asymptotic behaviour in different regions. In paper [1], the boundary value problems for the nonlinear Schrödinger equations on the half-line with homogeneous Robin boundary conditions were revisited using Bäcklund transformations. The results were illustrated by discussing several exact soliton solutions, which describe the soliton reflection at the boundary. ...
We study the initial-boundary value problem for the nonlinear fractional Schrödinger equation 0,~x>0; \\ u(x,0)={{u}_{0}}(x),~x>0,{{u}_{x}}(0,t)=h(t),~t>0. \end{array}\right. \end{eqnarray} >{ut+i(uxx+12π∫0∞sign(x−y)|x−y|12uy( y)dy)+i|u|2u=0, t>0, x>0;u(x,0)=u0(x), x>0,ux(0,t)=h(t), t>0.
We prove the global-in-time existence of solutions for a nonlinear fractional Schrödinger equation with inhomogeneous Neumann boundary conditions. We are also interested in the study of the asymptotic behaviour of the solutions.
... It was proved that the solution has different asymptotic behaviours in different regions. In Biondini & Bui [15], boundary-value problems for the NLS equations on the half-line with homogeneous Robin boundary conditions were revisited using Bäcklund transformations. The results were illustrated by discussing several exact soliton solutions, which described the soliton reflection at the boundary. ...
We consider the initial-boundary-value problem for the cubic nonlinear Schrödinger equation, formulated on a half-line with inhomogeneous Robin boundary data. We study traditionally important problems of the theory of nonlinear partial differential equations, such as the global-in-time existence of solutions to the initial-boundary-value problem and the asymptotic behaviour of solutions for large time.
In this paper, we study the initial-boundary-value problem for the cubic nonlinear Schrödinger equation, formulated on a half-line with different inhomogeneous boundaries data. We study the well-posedness in the case of Neumann and Robin condition in Sobolev space of low regularity. Also, we revisit, in a self-consistent way, some results concerning the Dirichlet condition.
In this topical review paper we provide a survey of classical and more recent results on the IST for one-dimensional scalar, vector and square matrix NLS systems on the line ( - ∞ < x < ∞ ) with certain physically relevant non-zero boundary conditions at space infinity, discuss some new developments and applications, and offer some perspectives about future directions on the subject.
We characterize initial value problems for the defocusing Manakov system (coupled two-component nonlinear Schrödinger equation) with nonzero background and well-defined spatial parity symmetry (i.e., when each of the components of the solution is either even or odd), corresponding to boundary value problems on the half line with Dirichlet or Neumann boundary conditions at the origin. We identify the symmetries of the eigenfunctions arising from the spatial parity of the solution, and we determine the corresponding symmetries of the scattering data (reflection coefficients, discrete spectrum and norming constants). All parity induced symmetries are found to be more complicated than in the scalar (i.e., one-component) case. In particular, we show that the discrete eigenvalues giving rise to dark solitons arise in symmetric quartets, and those giving rise to dark-bright solitons in symmetric octets. We also characterize the differences between the purely even or purely odd case (in which both components are either even or odd functions of x) and the “mixed parity” cases (in which one component is even while the other is odd). Finally, we show how, in each case, the spatial symmetry yields a constraint on the possible existence of self-symmetric eigenvalues, corresponding to stationary solitons, and we study the resulting behavior of solutions.
The general three-component nonlinear Schrödinger (gtc-NLS) equations are completely integrable and contain the self-focusing, defocusing and mixed cases, which are applied in many physical fields. In this paper, we would like to use the Fokas method to explore the initial-boundary value (IBV) problem for the gtc-NLS equations with a matrix Lax pair on a finite interval based on the inverse scattering transform. The solutions of the gtc-NLS equations can be expressed using the solution of a matrix Riemann–Hilbert (RH) problem constructed in the complex k-plane. The jump matrices of the RH problem can be explicitly found in terms of three spectral functions related to the initial data, and the Dirichlet–Neumann boundary data, respectively. The global relation between the distinct spectral functions is also proposed to derive two distinct but equivalent types of representations of the Dirichlet–Neumann boundary value problems. Particularly, the relevant formulae for the boundary value problems on the finite interval can generate ones on the half-line as the length of the interval closes to infinity. Finally, we also analyse the linearizable boundary conditions for the Gel’fand–Levitan–Marchenko representation. These results will be useful to further study the solution properties of the IBV problem of the gtc-NLS system by using the Deift–Zhou’s nonlinear steepest descent method and some numerical methods.
In this study, an efficient fourth-order conservative explicit numerical scheme using method of lines is developed to simulate different scenarios of soliton interactions and reflections for a (2 + 1)-dimensional coupled nonlinear Schrödinger (CNLS) system. The fourth-order Runge–Kutta technique is applied as a time integrator to the resulting ordinary differential system. Both integrable and nonintegrable cases of the CNLS system are considered. A condition for the scheme to be stable is deduced with the aid of von Neumann stability analysis. Several numerical experiments have been carried out to exhibit the reliability of the scheme in capturing and understanding the interesting phenomenon of elastic and inelastic soliton collisions/reflections related to many nonlinear evolution equations. The ability of the scheme to preserve the conserved invariants in long terms confirms its accuracy and stability. New results associated with interactions and reflections of soliton waves are obtained.
We consider the stochastic nonlinear Schrödinger equations on the half-line with mixed boundary conditions containing Brownian noise. The main novelty of this work is a convenient framework for the analysis of such problems. Our approach allows us to establish the global existence and uniqueness of a solution to initial-boundary value problem with values in H1.
We construct multi-soliton solutions of the n -component vector nonlinear Schrödinger equation on the half-line subject to two classes of integrable boundary conditions (BCs): the homogeneous Robin BCs and the mixed Neumann/Dirichlet BCs. The construction is based on the so-called dressing the boundary , which generates soliton solutions by preserving the integrable BCs at each step of the Darboux-dressing process. Under the Robin BCs, examples, including boundary-bound solitons, are explicitly derived; under the mixed Neumann/Dirichlet BCs, the boundary can act as a polarizer that tunes different components of the vector solitons. Connection of our construction to the inverse scattering transform is also provided.
We consider the inhomogeneous mixed initial-boundary value problem for the nonlinear multidimensional Schrödinger equation, formulated on upper right-quarter plane. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time. Also we are interested in the study of the influence of the mixta boundary data on the asymptotic behavior of solutions. Our approach to get well-posedness of nonlinear problems is based on the studying a linear theory and then using the fixed point argument. To get a linear theory for the multidimensional model we propose general method based on Laplace approach and theory Cauchy type integral equations. To get smooth solutions in L∞ we modify a method based on the factorization for the free Schrödinger evolution group. The advantage of our approach is that it can also be applied to non-integrable equations and arbitrary boundary conditions. This approach is new and it is not standard. We believe that the results of this paper could be applicable to study a wide class of dissipative multidimensional nonlinear equations by the use of techniques of nonlinear analysis.
Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schrödinger equation on the half‐line. The integrable BCs at the origin are represented by constraints of the Lax pair, and our method lies on dressing the Lax pair by preserving those constraints in the Darboux‐dressing process. The method is applied to two classes of solutions: solitons vanishing at infinity and self‐modulated solitons on a constant background. Half‐line solitons in both cases are explicitly computed. In particular, the boundary‐bound solitons, which are static solitons bounded at the origin, are also constructed. We give a natural inverse scattering transform interpretation of the method as evolution of the scattering data determined by the integrable BCs in space.
We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schrödinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques
We provide a direct method for constructing soliton solutions of the sine-Gordon equation in the laboratory coordinates on the half line in the presence of integrable boundary conditions derived in Sklyanin’s classical work (Sklyanin, 1987). Explicit examples, including boundary bound states, are presented.
This article is a continuation of the study in [5], where we proved the existence of solutions, global in time, for the initial-boundary value problem (Formula Presented) where is the module-fractional derivative operator defined by the modified Riesz Potential (Formula Presented) Here, we study the asymptotic behavior of the solution.
Boundary value problems for integrable nonlinear differential equations can be analyzed via the Fokas method. In this paper, this method is employed in order to study initial–boundary value problems of the general coupled nonlinear Schrödinger equation formulated on the finite interval with Lax pairs. The solution can be written in terms of the solution of a Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions , , and . The associated general Dirichlet to Neumann map is also analyzed via the global relation. It is interesting that the relevant formulas can be reduced to the analogous formulas derived for boundary value problems formulated on the half-line in the limit when the length of the interval tends to infinity. It is shown that the formulas characterizing the Dirichlet to Neumann map coincide with the analogous formulas obtained via a Gelfand–Levitan–Marchenko representation.
We present an approach for analyzing initial-boundary value problems which is
formulated on the finite interval (, where L is a positive
constant) for integrable equations whose Lax pairs involve
matrices. Boundary value problems for integrable nonlinear evolution PDEs can
be analyzed by the unified method introduced by Fokas and developed by him and
his collaborators. In this paper, we show that the solution can be expressed in
terms of the solution of a Riemann-Hilbert problem. The relevant
jump matrices are explicitly given in terms of the three matrix-value spectral
functions s(k),S(k) and , which in turn are defined in terms of the
initial values, boundary values at x=0 and boundary values at x=L,
respectively. However, these spectral functions are not independent, they
satisfy a global relation. Here, we show that the characterization of the
unknown boundary values in terms of the given initial and boundary data is
explicitly described for a nonlinear evolution PDE defined on the interval.
Also, we show that in the limit when the length of the interval tends to
infity, the relevant formulas reduce to the analogous formulas obtained for the
case of boundary value problems formulated on the half-line.
We consider the inhomogeneous Neumann initial-boundary value problem for the nonlinear Schrodinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.
We briefly investigated the well-known correlation of trihalomethanes, present in fresh water, with cancer hazard in humans. A transient alternative method of chemical simulation using Backlund Transformations and Quantum Mechanics is presented. Finally, the method was applied to simulate the interaction between Trichloridemethane and Alanine - as well as its amino and carboxyl groups.
Over the past thirty years significant progress has been made in the investigation of nonlinear waves--including "soliton equations", a class of nonlinear wave equations that arise frequently in such areas as nonlinear optics, fluid dynamics, and statistical physics. The broad interest in this field can be traced to understanding "solitons" and the associated development of a method of solution termed the inverse scattering transform (IST). The IST technique applies to continuous and discrete nonlinear Schrödinger (NLS) equations of scalar and vector type. This work presents a detailed mathematical study of the scattering theory, offers soliton solutions, and analyzes both scalar and vector soliton interactions. The authors provide advanced students and researchers with a thorough and self-contained presentation of the IST as applied to nonlinear Schrödinger systems.
We investigate the Manakov model or, more generally, the vector nonlinear Schrödinger equation on the half-line. Using a Bäcklund transformation method, two classes of integrable boundary conditions are derived: mixed Neumann/Dirichlet and Robin boundary conditions. Integrability is shown by constructing a generating function for the conserved quantities. We apply a nonlinear mirror image technique to construct the inverse scattering method with these boundary conditions. The important feature in the reconstruction formula for the fields is the symmetry property of the scattering data emerging from the presence of the boundary. Particular attention is paid to the discrete spectrum. An interesting phenomenon of transmission between the components of a vector soliton interacting with the boundary is demonstrated. This is specific to the vector nature of the model and is absent in the scalar case. For one-soliton solutions, we show that the boundary can be used to make certain components of the incoming soliton vanishingly small. This is reminiscent of the phenomenon of light polarization by reflection.
We consider the initial boundary value problem for the focusing nonlinear Schrödinger equation in the quarter plane x < 0, t < 0 in the case of decaying initial data (for t =0, as x →+∞) and the Robin boundary condition at x=0. We revisit the approach based on the simultaneous spectral analysis of the Lax pair equations and show that the method can be implemented without any a priori assumptions on the long-time behaviour of the boundary values. © 2012 The Author(s) Published by the Royal Society. All rights reserved.
We present an approach for analyzing initial-boundary value problems for
integrable equations whose Lax pairs involve matrices. Whereas
initial value problems for integrable equations can be analyzed by means of the
classical Inverse Scattering Transform (IST), the presence of a boundary
presents new challenges. Over the last fifteen years, an extension of the IST
formalism developed by Fokas and his collaborators has been successful in
analyzing boundary value problems for several of the most important integrable
equations with Lax pairs, such as the Korteweg-de Vries, the
nonlinear Schr\"odinger, and the sine-Gordon equations. In this paper, we
extend these ideas to the case of equations with Lax pairs involving matrices.
The problem of searching boundary value problems for soliton equations
consistent with the integrability property is discussed. A method of describing
integrals of motion for the integrable initial boundary value problems for the
Kadomtsev-Petviashvili equation is suggested via Green identity.
Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based on the elimination of the unknown boundary values by proper restrictions of the functional space and of the spectral variable complex domain. Illustrative examples include the linear Schroedinger equation on compact and semicompact n-dimensional domains and the nonlinear Schroedinger equation on the semiline.
The Nonlinear Schrodinger Equation (NS Model).- Zero Curvature Representation.- The Riemann Problem.- The Hamiltonian Formulation.- General Theory of Integrable Evolution Equations.- Basic Examples and Their General Properties.- Fundamental Continuous Models.- Fundamental Models on the Lattice.- Lie-Algebraic Approach to the Classification and Analysis of Integrable Models.- Conclusion.- Conclusion.
This book presents a new approach to analyzing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, a method that the author first introduced in 1997 and which is based on ideas of the inverse scattering transform. This method is unique in also yielding novel integral representations for the explicit solution of linear boundary value problems, which include such classical problems as the heat equation on a finite interval and the Helmholtz equation in the interior of an equilateral triangle. The author s thorough introduction allows the interested reader to quickly assimilate the essential results of the book, avoiding many computational details. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated radon transform and the Dirichlet-to-Neumann map for a moving boundary; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author s new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions. Audience: A Unified Approach to Boundary Value Problems is appropriate for courses in boundary value problems at the advanced undergraduate and first-year graduate levels. Applied mathematicians, engineers, theoretical physicists, mathematical biologists, and other scholars who use PDEs will also find the book valuable. Contents: Preface; Introduction; Chapter 1: Evolution Equations on the Half-Line; Chapter 2: Evolution Equations on the Finite Interval; Chapter 3: Asymptotics and a Novel Numerical Technique; Chapter 4: From PDEs to Classical Transforms; Chapter 5: Riemann Hilbert and d-Bar Problems; Chapter 6: The Fourier Transform and Its Variations; Chapter 7: The Inversion of the Attenuated Radon Transform and Medical Imaging; Chapter 8: The Dirichlet to Neumann Map for a Moving Boundary; Chapter 9: Divergence Formulation, the Global Relation, and Lax Pairs; Chapter 10: Rederivation of the Integral Representations on the Half-Line and the Finite Interval; Chapter 11: The Basic Elliptic PDEs in a Polygonal Domain; Chapter 12: The New Transform Method for Elliptic PDEs in Simple Polygonal Domains; Chapter 13: Formulation of Riemann Hilbert Problems; Chapter 14: A Collocation Method in the Fourier Plane; Chapter 15: From Linear to Integrable Nonlinear PDEs; Chapter 16: Nonlinear Integrable PDEs on the Half-Line; Chapter 17: Linearizable Boundary Conditions; Chapter 18: The Generalized Dirichlet to Neumann Map; Chapter 19: Asymptotics of Oscillatory Riemann Hilbert Problems; Epilogue; Bibliography; Index.
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 present a method to solve initial-boundary-value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A S Fokas to solve initial-boundary-value problems for linear and integrable nonlinear partial differential equations via an extension of the inverse scattering transform. The method takes advantage of the Lax pair formulation for both linear and nonlinear equations, and is based on the simultaneous spectral analysis of both parts of the Lax pair. A key role is also played by the global algebraic relation that couples all known and unknown boundary values. Even though additional technical complications arise in discrete problems compared to continuum ones, we show that a similar approach can also solve initial-boundary-value problems for linear and integrable nonlinear differential-difference equations. We demonstrate the method by solving initial-boundary-value problems for the discrete analogue of both the linear and the nonlinear Schrödinger equations, comparing the solution to those of the corresponding continuum problems. In the linear case we also explicitly discuss Robin-type boundary conditions not solvable by Fourier series. In the nonlinear case, we also identify the linearizable boundary conditions, we discuss the elimination of the unknown boundary datum, we obtain explicitly the linear and continuum limit of the solution, and we write the soliton solutions.
In the study of the equations on a finite segment that are integrable with the help of the method of inverse problem it is convenient to impose periodic boundary conditions or their variants [see V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. P. Pitaevskij, “Theory of solitons. The method of the inverse problem” (1980; Zbl 0598.35003) and L. D. Faddeev and L. A. Takhtadzhyan, “Hamiltonian methods in the theory of solitons” (1987; Zbl 0632.58004)]. Below we describe a new class of boundary conditions, compatible with complete integrability, for nonlinear equations that are integrable in the framework of ultralocal r-matrix scheme. The idea of the method, proposed here, has been suggested to the author by the article of I. V. Cherednik [Theor. Math. Phys. 61, 977-983 (1984); translation from Teor. Mat. Fiz. 61, No.1, 35-44 (1984; Zbl 0575.22021)].
Boundary initial value problems for integrable nonlinear equations are considered. Solutions of these problems with certain types of boundary conditions for the nonlinear Schrödinger equation and sine-Gordon equation are described in the scattering data terms.
A compendium of papers focusing on the development of, and the progress made with, the inverse scattering method in integrating important nonlinear differential wave equations is presented. An overview of the method is given, with derivations of the Kortweg-de Vries and nonlinear Schroedinger equations for certain physical problems. Periodic solutions of the Kortweg-de Vries equation are derived, and the construction of integrable systems and their solutions, symptotic solutions over large times, and the theory of the Kadomtsev-Petviashvili equation are examined.
Certain nonlinear evolution equations can be solved on the semi‐infinite interval by the method of inverse scattering. These equations are a subset of those which can be solved on the full interval. The equations have even dispersion relations when linearized, and are subject to appropriate homogeneous boundary conditions at the origin.
Assuming that the solution q(x, t) of the nonlinear Schrödinger equation on the half-line exists, it has been shown in Fokas (2002 Commun. Math. Phys. 230 1–39) that q(x, t) can be represented in terms of the solution of a matrix Riemann–Hilbert (RH) problem formulated in the complex k-plane. The jump matrix of this RH problem has explicit x, t dependence and it is defined in terms of the scalar functions {a(k), b(k), A(k), B(k)} referred to as spectral functions. The functions a(k) and b(k) are defined in terms of q0(x) = q(x,0), while the functions A(k) and B(k) are defined in terms of g0(t) = q(0,t) and g1(t) = qx(0,t). The spectral functions are not independent but they satisfy an algebraic global relation. Here we first prove that if there exist spectral functions satisfying this global relation, then the function q(x, t) defined in terms of the above RH problem exists globally and solves the nonlinear Schrödinger equation, and furthermore q(x, 0) = q0(x), q(0, t) = g0(t) and qx(0, t) = g1(t). We then show that, given appropriate initial and boundary conditions, it is possible to construct such spectral functions through the solution of a nonlinear Volterra integral equation whose solution exists globally. We also show that for a particular class of boundary conditions it is possible to bypass this nonlinear equation and to compute the spectral functions using only the algebraic manipulation of the global relation; thus for this particular class of boundary conditions, which we call linearizable, the problem on the half-line can be solved as effectively as the problem on the line. An example of a linearizable boundary condition is qx(0, t) − ρq(0, t) = 0 where ρ is a real constant.
The integrable initial boundary value problem on a semi-line for the nonlinear Schrodinger equation is considered. It is shown that by means of the Backlund transformation this problem can be reduced to the well known Cauchy problem for the same equation on the line.
The initial-boundary value problem on a semiline for the nonlinear Schrodinger or sine-Gordon equation is reduced to the problem on a line with special symmetry conditions. These conditions can be written in terms of scattering data in the framework of the inverse scattering transform. The reduction is done by means of a Backlund transform.
We solve the initial-boundary value problem (IBVP) for the Ablowitz–Ladik system on the natural numbers with certain linearizable boundary conditions. We do so by employing a nonlinear method of images, namely, by extending the scattering potential to all integers in such a way that the extended potential satisfies certain symmetry relations. Using these extensions and the solution of the initial value problem (IVP), we then characterize the symmetries of the discrete spectrum of the scattering problem, and we show that discrete eigenvalues in the IBVP appear in octets, as opposed to quartets in the IVP. Furthermore, we derive explicit relations between the norming constants associated with symmetric eigenvalues, and we identify a new kind of linearizable IBVP. Finally, we characterize the soliton solutions of these IBVPs, which describe the soliton reflection at the boundary of the lattice.
We consider the one-dimensional focusing nonlinear Schrödinger equation (NLS) with a delta potential and even initial data.
The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions
at the origin. We follow the method of Bikbaev and Tarasov which utilizes a Bäcklund transformation to extend the solution
on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary
1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann–Hilbert
problems introduced by Deift and Zhou. Our work strengthens, and extends, the earlier work on the problem by Holmer and Zworski.
Mathematics Subject Classification (2000)35Q15–35Q55
We present a method for studying initial-boundary value problems associated with integrable nonlinear evolution equations. For concreteness we consider the nonlinear Schrödinger equation in the variable q(x,t), x in [0,∞), with a mixed boundary condition, i.e. qx(0,t)+αq(0,t) is given (α is an arbitrary real constant), q(x,t) can be obtained by solving a linear integral equation uniquely defined in terms of appropriate scattering data. These data satisfy a single nonlinear integrodifferential equation uniquely defined in terms of the boundary condition. For the special case of a homogeneous boundary condition, the scattering data are found in closed form.
We characterize the soliton solutions of the nonlinear Schroedinger equation on the half line with linearizable boundary conditions. Using an extension of the solution to the whole line and the corresponding symmetries of the scattering data, we identify the properties of the discrete spectrum of the scattering problem. We show that discrete eigenvalues appear in quartets (as opposed to pairs in the initial value problem), and we obtain explicit relations for the norming constants associated to symmetric eigenvalues. The apparent reflection of each soliton at the boundary of the spatial domain is due to the presence of a "mirror" soliton, with equal amplitude and opposite velocity, located beyond the boundary. We then calculate the position shift of the physical solitons as a result of the nonlinear reflection. These results provide a nonlinear analogue of the method of images that is used to solve boundary value problems in electrostatics.
Unified Transform Method for Boundary Value Problems. SIAM, Philadelphia
- A Fokas
A S FOKAS, A Unified Transform Method for Boundary Value Problems. SIAM, Philadelphia, 2008. STATE UNIVERSITY OF NEW YORK (Received March 19, 2012)
Hamiltonian Methods in the Theory of Solitons
- L D Faddeev
- L A Takhtajan
L. D. FADDEEV, and L. A. TAKHTAJAN, Hamiltonian Methods in the Theory of Solitons,
Springer, Berlin, 1987.
A Unified Transform Method for Boundary Value Problems
- A S Fokas
A. S. FOKAS, A Unified Transform Method for Boundary Value Problems. SIAM,
Philadelphia, 2008.
STATE UNIVERSITY OF NEW YORK
(Received March 19, 2012)
- Ablowitz
Initial boundary value problem for the focusing NLS equation with Robin boundary conditions: half-line approach
- A Ritsanddshepelsky