反散射变换和正规Riemann-Hilbert问题

本文指出,非线性演化方程Lax表示的反散射交换可以化为关于亚纯函数的正规Riemann-Hilbert问题,并导出了相应的积分方程,后者在实质上与Gel’fand-Levitan-Marchenko方程等价。本文进而导出Lax表示的不同基本解组之间的Darboux-B?cklund变换所满足的正规Riemann-Hilbert问题。与反散射变换直接联系的正规Riemann-Hilbert问题是其特殊形式。 关键词：     

An Inverse Scattering Transformation for the Modified Nonlinear Schrijdinger Equation

A new Lax pair of the modified nonlinear Schrödinger equation is introduced in terme of the variable of the Fourier transform λ. The Lax pair has no usual symmetries between 12 and 21 elements and avoids the factor λ^1/2. The basic equation of inverse scattering transformation is deduced in the Zakharov-Shabat form as well as in the Marchenko form.     

Riemann?CHilbert Approach to the Elastodynamic Equation: Part I

We develop the Riemann?CHilbert (RH) approach to scattering problems in elastic media. The approach is based on the RH method introduced in the 1990s by Fokas (A unified approach to boundary value problems, CBMS-SIAM, 2008) for studying boundary problems for linear and integrable nonlinear PDEs. A suitable Lax pair formulation of the elastodynamic equation is obtained. The integral representations derived from this Lax pair are applied to Rayleigh wave propagation in an elastic half space and quarter space. The latter problem is reduced to the analysis of a certain underdetermined RH problem. We show that the problem can be re-formulated as a well-determined vector Riemann?CHilbert problem with a shift posed on a torus.     

Reduced Maxwell-Bloch equations with anisotropic dipole momentum

We develop the inverse scattering transform for the recently found integrable system of reduced Maxwell-Bloch equations with two components of polarization and with an anisotropic dipole momentum. The model describes few-cycle pulses of optical or other field propagations. We find that the existence of a nontrivial group of symmetry of the corresponding Lax pair leads to a particular form of the inverse scattering transform equations. We show that solutions can be expressed in terms of the solution of a matrix Riemann-Hilbert problem formulated for the complex plane with a nontrivial group of automorphisms.     

On a spectral analysis of scattering data for the Camassa-Holm equation

Physical details of the Camassa–Holm (CH) equation that are difficult to obtain in space-time simulation can be explored by solving the Lax pair equations within the direct and inverse scattering analysis context. In this spectral analysis of the completely integrable CH equation we focus solely on the direct scattering analysis of the initial condition defined in the physical space coordinate through the time-independent Lax equation. Both of the continuous and discrete spectrum cases for the initial condition under current investigation are analytically derived. The scattering data derived from the direct scattering transform for non-reflectionless case are also discussed in detail in spectral domain from the physical viewpoint.     

Darboux transformation and soliton solutions of a nonlocal Hirota equation

Starting from local coupled Hirota equations,we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem.The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair.By Lax pair,an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found.The solutions with specific properties are distinct from those of the local Hirota equation.In order to further describe the properties and the dynamic features of the solutions explicitly,several kinds of graphs are depicted.     

Prolongation structure of a new integrable system

We have obtained the inverse scattering equations associated with a new pair of coupled nonlinear evolution equations in two dimensions. The spectral parameter is introduced by invoking the invariance of the equation set, and imposing those on the Lax pair.     

Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.     

Connections defining representations of zero curvature and the solitons of sine-Gordon and Korteweg-de Vries equations

It is claimed that solutions of travelling-wave type (and, in particular, soliton solutions) of partial differential equations can be created by using connections defining representations of zero curvature. In this paper, we construct solitons of the sine-Gordon and Korteweg-de Vries equations. By previous results of the author, the connections defining representations of zero curvature for a given differential equation generate Bäcklund transformations for this equation. It can be shown that the well-known Lax system (the so-called Lax pair) for the Korteweg-de Vries equation is a special case of a Bäcklund system (i.e., the system of partial differential equations defining a Bäcklund transformation). Note that the creation of solitons by means of the inverse scattering method is in fact a creation of solitons by means of the Lax system (without using connections defining the representations of zero curvature from the very beginning). Moreover, the inverse scattering method is essentially more labor-consuming than the method suggested in the present paper. Further, it is not required to involve any physical notions when using the suggested method. In the final section of the paper, we consider the so-called 2-soliton solutions of sine-Gordon and Korteweg — de Vries equations. Here we systematically use the invariant analytic method developed by G. F. Laptev, which is well-known in differential geometry under the title of Cartan-Laptev method.     

The Elliptic Sinh-Gordon Equation in the Quarter Plane

We study the elliptic sinh-Gordon equation formulated in the quarter plane by using the so-called Fokas method, which is a signi?cant extension of the inverse scattering transform for the boundary value problems. The method is based on the simultaneous spectral analysis for both parts of the Lax pair and the global algebraic relation that involves all boundary values. In this paper, we address the existence theorem for the elliptic sinh-Gordon equation posed in the quarter plane under the assumption that the boundary values satisfy the global relation. We also present the formal representation of the solution in terms of the unique solution of the matrix Riemann- Hilbert problem de?ned by the spectral functions.     

Inverse scattering method and vector higher order non-linear Schrödinger equation

A generalized inverse scattering method has been developed for arbitrary n-dimensional Lax equations. Subsequently, the method has been used to obtain N-soliton solutions of a vector higher order non-linear Schrödinger equation, proposed by us. It has been shown that under a suitable reduction, the vector higher order non-linear Schrödinger equation reduces to the higher order non-linear Schrödinger equation. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. Gauge equivalence is shown between the vector higher order non-linear Schrödinger equation and the generalized Landau–Lifshitz equation and the Lax pair for the latter equation has also been constructed in terms of the spin field, establishing direct integrability of the spin system.     

Orthogonal polynomial approach to discrete Lax pairs for initial boundary-value problems of the QD algorithm

Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials corresponding to a given linear form. Shifts on the linear form give rise to adjacent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover the discrete-time Toda chain equations of Hirota and of Suris. This approach allows us to derive a Bäcklund transform that relates these two different discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity confinement property is discussed as well.     

Gauge Transformation and Generalized Nonlocal Symmetries of the Korteweg-de Vries Equation

After extending the usual Lax pair of the Korteweg-de Vries (KdV) equation to a generalized form by using a gauge transformation, an adjoint Lax pair of the KdV equation is introduced. With the help of the spectral functions of the Lax pair and the adjoint Lax pair, a new nonlocal seed symmetry (which is gauge-invariant) is found and then a set of new infinitely many generalized nonlocal symmetries are obtained after establishing a general symmetry theory for an arbitrary nonlinear system.     

Inverse scattering transforms of the inhomogeneous fifth-order nonlinear Schrödinger equation with zero/nonzero boundary conditions

The present work studies the inverse scattering transforms (IST) of the inhomogeneous fifth-order nonlinear Schrödinger (NLS) equation with zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs). Firstly, the bound-state solitons of the inhomogeneous fifth-order NLS equation with ZBCs are derived by the residue theorem and the Laurent’s series for the first time. Then, by combining with the robust IST, the Riemann-Hilbert (RH) problem of the inhomogeneous fifth-order NLS equation with NZBCs is revealed. Furthermore, based on the resulting RH problem, some new rogue wave solutions of the inhomogeneous fifth-order NLS equation are found by the Darboux transformation. Finally, some corresponding graphs are given by selecting appropriate parameters to further analyze the unreported dynamic characteristics of the corresponding solutions.     

Extremely short vector solitons under the conditions of conical refraction

Propagation of extremely short electromagnetic pulses in a biaxial crystal under the conditions of conical refraction has been considered. The system of wave equations taking into account the dispersion contribution of the crystal lattice ions to the polarization response of the medium and a nonlinearity of the polarization response of electrons has been derived. It has been shown that under certain conditions this system can be reduced to an equation which is integrable by means of the inverse scattering transformation method. The proper Lax pair has been found. Physical analysis of the steady-state pulse solution of the system of wave equations has been performed.     

ON THE PROLONGATION STRUCTURE OF ERNST EQUATION

An SL(2R) ×R¹(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Bäcklund transformations is pfesented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neugebauer transformations and other available Bäcklund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbertproblem (HHP), would be covered by this condition.     

Integrability of extended (2+1)-dimensional shallow water wave equation with Bell polynomials

We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.     

The Matrix Kadomtsev–Petviashvili Equation as a Source of Integrable Nonlinear Equations

Abstract

A new integrable class of Davey–Stewartson type systems of nonlinear partial differential equations (NPDEs) in 2+1 dimensions is derived from the matrix Kadomtsev– Petviashvili equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatio-temporal rescaling. The integrability by the inverse scattering method is explicitly demonstrated, by applying the reduction technique also to the Lax pair of the starting matrix equation and thereby obtaining the Lax pair for the new class of systems of equations. The characteristics of the reduction method suggest that the new systems are likely to be of applicative relevance. A reduction to a system of two interacting complex fields is briefly described.     

Solutons of a nonisospectral and variable coefficient Korteweg-de Vries equation

A new type of KdV equation with a nonisospectral Lax pair as well as variable coefficients is introduced. Its Lax pair is shown to be invariant under the Crum transformation. This leads to a Bäcklund transformation for the KdV equation and, hence, a method for solutions via an associated nonisospectral variable coefficient MKdV equation. Three generations of solutions are given. The 1-soliton solution shares the novel phenomenology associated with the boomeron, trappon, and zoomeron of Calogero and Degasperis.     

An extension of the modified Sawada—Kotera equation and conservation laws

Based on the modified Sawada-Kotera equation, we introduce a 3 × 3 matrix spectral problem with two potentials and derive a hierarchy of new nonlinear evolution equations. The second member in the hierarchy is a generalization of the modified Sawada-Kotera equation, by which a Lax pair of the modified Sawada-Kotera equation is obtained. With the help of the Miura transformation, explicit solutions of the Sawada-Kotera equation, the Kaup-Kupershmidt equation, and the modified Sawada-Kotera equation are given. Moreover, infinite sequences of conserved quantities of the first two nonlinear evolution equations in the hierarchy and the modified Sawada-Kotera equation are constructed with the aid of their Lax pairs.