INVERSE SCATTER1 NG TRANSFORM AND REGULAR RIEMANN-HILBERT PROBLEM

It is shown that the inverse scattering transform method for solving the Lax pair of given nonlinear evolution equation can be reduced to a kind of Riemann-Hilbert (RH) problem of meromorphic functions with respect to the complex spectral parameter. The RH problem is generally regular no matter solitons are involved or not. The linear singular integral equation associated with the RH problem has been derived, which is essential1y equivalent to the Gel fand-Levitan-Marchenko equation.Furthermore, the regtllar RH problem satisfied by the Sacklund transformation from a fundamental solution set of the eigenvalue equations of Lax pair to a new set has Fen given as well. The RH problem reduced from the inverse scattering transform is in fact a special case of that satisfied by the Backlund transformation.     

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：     

The integration algorithm of Lax equation for both generic Lax matrices and generic initial conditions

Several physical applications of Lax equation require its general solution for generic Lax matrices and generic not necessarily diagonalizable initial conditions. In the present paper we complete the analysis started in [arXiv:0903.3771] on the integration of Lax equations with both generic Lax operators and generic initial conditions. We present a complete general integration formula holding true for any (diagonalizable or non-diagonalizable) initial Lax matrix and give an original rigorous mathematical proof of its validity relying on no previously published results.     

An Integrable Symplectic Map of a Differential-Difference Hierarchy

By choosing a discrete matrix spectral problem, a hierarchy of integrable differential-difference equations is derived from the discrete zero curvature equation, and the Hamiltonian structures are built. Through a higher-order Bargmann symmetry constraint, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs, which we obtained are respectively nonlinearized into a new integrable symplectic map and a finite-dimensional integrable Hamiltonian system in Liouville sense.     

Two hierarchies of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformation

By considering a new discrete isospectral eigenvalue problem, two hierarchies of integrable positive and negative lattice models are derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And, each equation in the resulting hierarchies is integrable in Liouville sense. Moreover, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.     

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.     

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.     

Symmetry Reduction of (2+1)-Dimensional Lax–Kadomtsev–Petviashvili Equation

The Lax–Kadomtsev–Petviashvili equation is derived from the Lax fifth order equation, which is an important mathematical model in fluid physics and quantum field theory. Symmetry reductions of the Lax–Kadomtsev–Petviashvili equation are studied by the means of the Clarkson–Kruskal direct method and the corresponding reduction equations are solved directly with arbitrary constants and functions.     

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.     

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.     

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.     

Lax可积系统的有限变换对称群及严格解

本文用一种简单直接的方法研究了非线性物理系统Davey-Stewartson方程及（２＋１）维的Camassa-Holm方程，并给出了它们的有限变换对称群及对称。     

Lump and lump-soliton interaction solutions for an integrable variable coefficient Kadomtsev-Petviashvili equation

For a variable coefficient Kadomtsev-Petviashvili (KP) equation the Lax pair as well as conjugate Lax pair are derived from the Painleve analysis.The N-fold binary Darboux transformation is presented in a compact form.As an application,the multi-lump,higher-order lump and general lump-soliton interaction solutions for the variable coefficient KP equation are obtained.Typical lump structures with amplitudes exponentially decaying to zero as the time tends to infinity and interactions between one lump and one soliton are shown.     

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.     

Integrable Hierarchy Covering the Lattice Burgers Equation in Fluid Mechanics:N-fold Darboux Transformation and Conservation Laws

Burgers-type equations can describe some phenomena in fluids,plasmas,gas dynamics,traffic,etc.In this paper,an integrable hierarchy covering the lattice Burgers equation is derived from a discrete spectral problem.N-fold Darboux transformation(DT) and conservation laws for the lattice Burgers equation are constructed based on its Lax pair.N-soliton solutions in the form of Vandermonde-like determinant are derived via the resulting DT with symbolic computation,structures of which are shown graphically.Coexistence of the elastic-inelastic interaction among the three solitons is firstly reported for the lattice Burgers equation,even if the similar phenomenon for certern continuous systems is known.Results in this paper might be helpful for understanding some ecological problems describing the evolution of competing species and the propagation of nonlinear waves in fluids.     

Integrable Hierarchy Covering the Lattice Burgers Equation in Fluid Mechanics: N-fold Darboux Transformation and Conservation Laws

Burgers-type equations can describe some phenomena in fluids, plasmas, gas dynamics, traffic, etc. In this paper, an integrable hierarchy covering the lattice Burgers equation is derived from a discrete spectral problem. N-fold Darboux transformation (DT) and conservation laws for the lattice Burgers equation are constructed based on its Lax pair. N-soliton solutions in the form of Vandermonde-like determinant are derived via the resulting DT with symbolic computation, structures of which are shown graphically. Coexistence of the elastic-inelastic interaction among the three solitons is firstly reported for the lattice Burgers equation, even if the similar phenomenon for certern continuous systems is known. Results in this paper might be helpful for understanding some ecological problems describing the evolution of competing species and the propagation of nonlinear waves in fluids.     

Complete integrability of the asymmetric chiral O(3)-field equation in a class of rapidly decreasing functions

The direct and inverse scattering problem for the asymmetric chiral O(3)-field equation is investigated by the Gel'fand-Levitan approach. The same results for the Landau-Lifshitz equation are presented. The canonical variables are derived and the Hamiltonian perturbation theory of these equations is constructed. It is not difficult to apply the methods of the paper to any L-operator with spectral parameter on a curve of genus g > 0.     

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 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.     

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.