Novel soliton structures are constructed for the Fokas-Lenells equation. In so doing, and after discussing the stability of continuous waves, a multiple scales perturbation theory is used to reduce the equation to a Korteweg-de Vries system whose relative soliton solution gives rise to intricate (and rather unexpected) solutions to the original system. Both the focusing and defocusing equations are considered and it is found that dark solitons may exist in both cases while in the focusing case antidark solitons are also possible. These findings are quite surprising as the relative nonlinear Schrödinger equation does not exhibit these solutions.
This work is devoted to an integrable generalization of the nonlinear Schrödinger equation proposed by Fokas and Lenells. I discuss the relationships between this equation and other integrable models. Using the reduction of the Fokas-Lenells equation to the already known ones I obtain the N-dark soliton solutions.
This paper focuses on the modulation instability, conservation laws and localized wave solutions of the generalized coupled Fokas-Lenells equation. Based on the theory of linear stability analysis, distribution pattern of modulation instability gain G in the (K,k) frequency plane is depicted, and the constraints for the existence of rogue waves are derived. Subsequently, we construct the infinitely many conservation laws for the generalized coupled Fokas-Lenells equation from the Riccati-type formulas of the Lax pair. In addition, the compact determinant expressions of the N-order localized wa…
In this paper, the Fokas-Lenells equations are investigated via bilinear approach. We bilinearize the unreduced Fokas-Lenells system, derive double Wronskian solutions, and then, by means of a reduction technique we obtain variety of solutions of the reduced equations. This enables us to have a full profile of solutions of the classical and nonlocal Fokas-Lenells equations. Some obtained solutions are illustrated based on asymptotic analysis. As a notable new result, we obtain solutions to the Fokas-Lenells equation, which are related to real discrete eigenvalues and not reported before in the…
This paper is dedicated to provide theta function representations of algebro-geometric solutions for the Fokas-Lenells (FL) hierarchy through studying an algebro-geometric initial value problem. Further, we reduce these solutions into $N$-dark solutions through the degeneration of associated Riemann surfaces.
In a previous study, we obtained the algebro-geometric solutions and $n$-dark solitons of Forkas-Lenells (FL) hierarchy using algebro-geometric method. In this paper, we construct physically relevant classes of solutions for FL hierarchy by studying the reality conditions for $q=\pm \bar{r}$ based on the idea of Vinikov's homological basis.
The multi-component Fokas-Lenells equation is considered. In particular, we present the multisoliton formulas for the system with plane-wave boundary conditions, as well as with mixed zero and plane-wave boundary conditions. A direct approach is employed to construct solutions, showing that for both boundary conditions, the multisoliton solutions have compact determinantal expressions.
In this paper, we firstly establish the multi-Hamiltonian structure and infinite many conservation laws for the vector Kaup-Newell hierarchy of the positive and negative orders. The first nontrivial negative flow corresponds to a coupled Fokas-Lenells equation. By constructing a generalized Darboux transformation and using a limiting process, all kinds of one-soliton solutions are constructed including the bright-dark soliton, the dark-anti-dark soliton and the breather-like solutions. Furthermore, multi-bright and multi-dark soliton solutions are derived and their asymptotic behaviors are inv…
In this article, we focus on the inverse scattering transformation for the Fokas-Lenells (FL) equation with nonzero boundary conditions via the Riemann-Hilbert (RH) approach. Based on the Lax pair of the FL equation, the analyticity and symmetry, asymptotic behavior of Jost solutions and scattering matrix are discussed in detail. With these results, we further present a generalized RH problem, from which a reconstruction formula between the solution of the FL equation and the Riemann-Hilbert problem is obtained. The N-soliton solutions of the FL equation is obtained via solving the RH problem.
We use the Deift-Zhou method to obtain, in the solitonless sector, the leading order asymptotic of the solution to the Cauchy problem of the Fokas-Lenells equation as $t\ra+\infty$ on the full-line.
Considering certain terms of the next asymptotic order beyond the nonlinear Schrodinger equation (NLS) equation, the Fokas-Lenells (FL) equation governed by the FL system arise as a model for nonlinear pulse propagation in optical fibers. The expressions of the q[n] and r[n] in the FL system are generated by n-fold Darboux transformation (DT). Further, a Taylor series expansion about the n-order breather solutions q[n] generated using DT by assuming periodic seed solutions under reduction can generate the n-order rogue waves of the FL equation explicitly with 2n+3 free parameters.
We develop a direct method of solution for finding the bright $N$-soliton solution of the Fokas-Lenells derivative nonlinear Schrödinger equation. The construction of the solution is performed by means of a purely algebraic procedure using an elementary theory of determinants and does not rely on the inverse scattering transform method. We present two different expressions of the solution both of which are expressed as a ratio of determinants. We then investigate the properties of the solutions and find several new features. Specifically, we derive the formula for the phase shift caused by the…
In this paper, we use the unified transform method to consider the initial-boundary value problem for the coupled Fokas-Lenells equations on the half-line, assuming that the solution $\{q(x,t),r(x,t)\}$ of the coupled Fokas-Lenells equations exists, we show that $\{q_x(x,t),r_x(x,t)\}$ can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $λ$. Thus, the solution $\{q(x,t),r(x,t)\}$ can be obtained by integration with respect to $x$.
The concept of the nonholonomic deformation formulated recently for the AKNS family is extended to the Kaup-Newell class. Applying this construction we discover a novel two-fold integrable hierarchy related to the deformed derivative nonlinear Schrödinger (DNLS) equation and found the exact soliton solutions exhibiting unusual accelerating motion for both its field and the perturbing functions. Extending the idea of deformation the integrable perturbation of the gauge related Chen-Lee-Liu DNLS equation is constructed together with its soliton solution. We show that, the recently proposed Lenel…
We study the Schwartz class of initial-boundary value (IBV) problems for the integrable Fokas-Lenells equation on the half-line via the Deift-Zhou's nonlinear descent method analysis of the corresponding Riemann-Hilbert problem such that the asymptotics of the Schwartz class of IBV problems as t\to\infty is presented.
Multi-component integrable generalizations of the Fokas-Lenells equation, associated with each irreducible Hermitian symmetric space are formulated. Description of the underlying structures associated to the integrability, such as the Lax representation and the bi-Hamiltonian formulation of the equations is provided. Two reductions are considered as well, one of which leads to a nonlocal integrable model. Examples with Hermitian symmetric spaces of all classical series of types A.III, BD.I, C.I and D.III are presented in details, as well as possibilities for further reductions in a general for…
Integrable motion of curves associated with the Fokas-Lenells equation and related spin system
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）；Cotton University（科顿大学）；Bharathidasan University（巴拉蒂达桑大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
The Fokas-Lenells (FL) equation arises as a model eqution which describes for nonlinear pulse propagation in optical fibers by retaining terms up to the next leading asymptotic order (in the leading asymptotic order the nonlinear Schrödinger (NLS) equation results). Here we present an explicit analytical representation for the rogue waves of the FL equation. This representation is constructed by deriving an appropriate Darboux transformation (DT) and utilizing a Taylor series expansion of the associated breather solution. when certain higher-order nonlinear effects are considered, the propagat…
In this work, the generalized $N$-component Fokas-Lenells(FL) equations, which have been studied by Guo and Ling (2012 J. Math. Phys. 53 (7) 073506) for $N=2$, are first investigated via Riemann-Hilbert(RH) approach. The main purpose of this is to study the soliton solutions of the coupled Fokas-Lenells(FL) equations for any positive integer $N$, which have more complex linear relationship than the analogues reported before. We first analyze the spectral analysis of the Lax pair associated with a $(N+1)\times (N+1)$ matrix spectral problem for the $N$-component FL equations. Then, a kind of RH…
In a previous study (Matsuno Y, J. Phys. A: Math. Theor. 45(2012)23202), we have developed a systematic method for obtaining the bright soliton solutions of the Fokas-Lenells derivative nonlinear Schrödinger equation (FL equation shortly) under vanishing boundary condition. In this paper, we apply the method to the FL equation with nonvanishing boundary condition. In particular, we deal with a more sophisticated problem on the dark soliton solutions with a plane wave boundary condition. We first derive the novel system of bilinear equations which is reduced from the FL equation through a depen…
We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation $$ u_{xt}+αβ^2u-2iαβu_x-αu_{xx}-iαβ^2|u|^2u_x=0 \label{cs} $$ with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at $k=0$. (ii) four stationary phase points will appear during asymptotic analysi…
On the global well-posedness for the nonlocal Fokas-Lenells equation with the weighted Sobolev initial data on the line
Zhongyuan University of Technology（中原工学院）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学学院国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Soliton Management for ultrashort pulse: dark and anti-dark solitons of Fokas-Lenells equation with a damping like perturbation and a gauge equivalent spin system
An integrable extension of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations is investigated in this paper.We will refer to this integrable extension as the (4+1)-dimensional Fokas equation. The determinant expressions of soliton, breather, rational, and semi-rational solutions of the (4 + 1)-dimensional Fokas equation are constructed based on the Hirota's bilinear method and the KP hierarchy reduction method. The complex dynamics of these new exact solutions are shown in both three-dimensional plots and two-dimensional contour plots. Interestingly, the patterns of obtained h…
We analyze a generalization of the sine-Gordon equation in laboratory coordinates on the half-line. Using the Fokas transform method for the analysis of initial-boundary value problems for integrable PDEs, we show that the solution $u(x,t)$ can be constructed from the initial and boundary values via the solution of a $2\times 2$-matrix Riemann-Hilbert problem.
It is known that the initial-boundary value problem for certain integrable partial differential equations (PDEs) on the half-line with integrable boundary conditions can be mapped to a special case of the Inverse Scattering Method (ISM) on the full-line. This can also be established within the so-called Unified Transform (UT) for initial-boundary value problems with linearizable boundary conditions. In this paper, we show a converse to this statement within the Ablowitz-Kaup-Newell-Segur (AKNS) scheme: the ISM on the full-line can be mapped to an initial-boundary value problem with linearizabl…
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 algebra…
We employ the Ablowitz-Ladik system as an illustrative example in order to demonstrate how to analyze initial-boundary value problems for integrable nonlinear differential-difference equations via the unified transform (Fokas method). In particular, we express the solutions of the integrable discrete nonlinear Schrödinger and integrable discrete modified Korteweg-de Vries equations in terms of the solutions of appropriate matrix Riemann-Hilbert problems. We also discuss in detail, for both the above discrete integrable equations, the associated global relations and the process of eliminating o…
We present an approach for analyzing initial-boundary value problems for integrable equations whose Lax pairs involve $3 \times 3$ 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 $2 \times 2$ Lax pairs, such as the Korteweg-de Vrie…
In this work, by using the Hirota bilinear method, we obtain one- and two-soliton solutions of integrable $(2+1)$-dimensional $3$-component Maccari system which is used as a model describing isolated waves localized in a very small part of space and related to very well-known systems like nonlinear Schrödinger, Fokas, and long wave resonance systems. We represent all local and Ablowitz-Musslimani type nonlocal reductions of this system and obtain new integrable systems. By the help of reduction formulas and soliton solutions of the $3$-component Maccari system, we obtain one- and two-soliton s…
Recently, an integrable system of coupled (2+1)-dimensional nonlinear Schrodinger (NLS) equations was introduced by Fokas (eq. (18) in Nonlinearity 29}, 319324 (2016)). Following this pattern, two integrable equations [eqs.2 and 3] with specific parity-time symmetry are introduced here, under different reduction conditions. For eq. 2, two kinds of periodic solutions are obtained analytically by means of the Hirota's bilinear method. In the long-wave limit, the two periodic solutions go over into rogue waves (RWs) and semi-rational solutions, respectively. The RWs have a line shape, while the s…
We present an approach for analyzing initial-boundary value problems which is formulated on the finite interval ($0\le x\le L$, where $L$ is a positive constant) for integrable equations whose Lax pairs involve $3\times 3$ 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 $3\times 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix…
We generalize earlier results of Fokas and Liu and find all locally analytic (1+1)-dimensional evolution equations of order $n$ that admit an $N$-shock type solution with $N\leq n+1$. To this end we develop a refinement of the technique from our earlier work (A. Sergyeyev, J. Phys. A: Math. Gen, 35 (2002), 7653--7660), where we completely characterized all (1+1)-dimensional evolution systems $\bi{u}_t=\bi{F}(x,t,\bi{u},\p\bi{u}/\p x,...,\p^n\bi{u}/\p x^n)$ that are conditionally invariant under a given generalized (Lie--Bäcklund) vector field $\bi{Q}(x,t,\bi{u},\p\bi{u}/\p x,...,\p^k\bi{u}/\p …
The main aim of this paper is to discuss recent results on the adaptation of the Fokas-Gel'fand procedure for constructing soliton surfaces in Lie algebras, which was originally derived for PDEs [Grundland, Post 2011], to the case of integrable ODEs admitting Lax representations. We give explicit forms of the $\g$-valued immersion functions based on conformal symmetries involving the spectral parameter, a gauge transformation of the wave function and generalized symmetries of the linear spectral problem. The procedure is applied to a symmetry reduction of the static $ϕ^4$-field equations leadi…
The main aim of this paper is to introduce a new version of the Fokas-Gel'fand formula for immersion of soliton surfaces in Lie algebras. The paper contains a detailed exposition of the technique for obtaining exact forms of 2D-surfaces associated with any solution of a given nonlinear ordinary differential equation (ODE) which can be written in zero-curvature form. That is, for any generalized symmetry of the zero-curvature condition of the associated integrable model, it is possible to construct soliton surfaces whose Gauss-Mainardi-Codazzi equations are equivalent to infinitesimal deformati…
