Integrability Properties of A Supersymmetric Coupled Dispersionless Integrable System

We study the integrability aspects of an N=1 supersymmetric coupled dispersionless (SUSY-CD) integrable system in detail. We present a superfield Lax representation of the SUSY-CD system by writing its (3×3)-matrix superfield Lax pair and show that the zero-curvature condition corresponds to the SUSY-CD system. From the fermionic superfield Lax representation, we obtain a set of coupled superfield Riccati equations that we further use to obtain an infinite set of superfield conserved currents. We investigate the Darboux transformation of the SUSY-CD system and use it to obtain multisoliton solutions of the system.     

A generalization of the coupled integrable dispersionless equations

The paper investigates an extension of the coupled integrable dispersionless equations, which describe the current‐fed string within an external magnetic field. By using the relation among the coupled integrable dispersionless equations, the sine‐Gordon equation and the two‐dimensional Toda lattice equation, we propose a generalized coupled integrable dispersionless system. N‐soliton solutions to the generalized system are presented in the Casorati determinant form with arbitrary parameters. By choosing real or complex parameters in the Casorati determinant, the properties of one‐soliton and two‐soliton solutions are investigated. It is shown that we can obtain solutions in soliton profile and breather profile. Copyright © 2016 John Wiley & Sons, Ltd.     

Darboux transformation and explicit solutions for two integrable equations

A new N-fold Darboux transformation for two integrable equations is constructed with the help of a gauge transformation for the spectral problem proposed by Qiao [Z.J. Qiao, Phys. Lett. A 192 (1994) 316-322]. By the Darboux transformation, explicit soliton and multi-soliton solutions for the two equations are obtained. In particular, soliton and complexiton solutions are shown through some figures.     

A coupled nonlinear Schrödinger type equation and its explicit solutions

A new coupled nonlinear Schrödinger type equation is proposed and is proved to be completely integrable. Based on the resulting Lax pair, a Darboux transformation for the coupled nonlinear Schrödinger type equation is derived with the aid of a gauge transformation between spectral problems. As an application, some explicit solutions of the coupled nonlinear Schrödinger type equation are obtained, including periodic and rational solutions.     

Explicit solutions to a coupled integrable lattice equation

A coupled integrable lattice equation is derived from a 4 × 4 matrix spectral problem, then with the help of a special Darboux matrix, explicit solutions of the aforementioned equation are given by means of gauge transformation between the Lax pair. Finally, the density profiles of these exact solutions are presented to illustrate these solutions.     

Explicit Solutions to an Integrable Lattice

In this paper, we aim to construct the Darboux transformation and explicit solutions for an integrable lattice introduced by Suris. Analysis of properties of the solutions shows that the obtained explicit solutions for this discrete integrable system possess new dynamical characters which are different from the ones of continuous integrable systems.     

Exact coherent structures for coupled integrable dispersionless equations

In this paper, the singular manifold method is applied to search coherent structures in an analytical form for the coupled integrable dispersionless equations. The Generalized solutions have been derived to the coupled integrable dispersionless equations, where the solutions are determined by the singular variable totally. With the aid of symbolic computation and plot representation of Maple, some coherent structures expressed in terms of new forms, such as solitoffs and breather lattice structures, have been illustrated by means of arbitrary functions in the analytical forms.     

COUPLED NONLOCAL NONLINEAR SCHR?DINGER EQUATION AND N-SOLITON SOLUTION FORMULA WITH DARBOUX TRANSFORMATION

Ablowitz and Musslimani proposed some new nonlocal nonlinear integrable equations including the nonlocal integrable nonlinear Schr?dinger equation. In this paper, we investigate the Darboux transformation of coupled nonlocal nonlinear Schr?dinger(CNNLS) equation with a spectral problem. Starting from a special Lax pairs, the CNNLS equation is constructed. Then, we obtain the one-, two-and N-soliton solution formulas of the CNNLS equation with N-fold Darboux transformation. Based on the obtained solutions, the propagation and interaction structures of these multi-solitons are shown, the evolution structures of the one-dark and one-bright solitons are exhibited with N = 1,and the overtaking elastic interactions among the two-dark and two-bright solitons are considered with N = 2. The obtained results are different from those of the solutions of the local nonlinear equations. Some different propagation phenomena can also be produced through manipulating multi-soliton waves.The results in this paper might be helpful for understanding some physical phenomena described in plasmas.     

Integrable discretization and numerical simulations of the generalized coupled integrable dispersionless equations

In this paper, we study the generalized coupled integrable dispersionless (GCID) equations and construct two integrable discrete analogues including a semi-discrete system and a full-discrete one. The results are based on the relations among the GCID equations, the sine-Gordon equation and the two-dimensional Toda lattice equation. We also present the N-soliton solutions to the semi-discrete and fully discrete systems in the form of Casorati determinant. In the continuous limit, we show that the fully discrete GCID equations converge to the semi-discrete GCID equations, then further to the continuous GCID equations. By using the integrable semi-discrete system, we design two numerical schemes to the GCID equations and carry out several numerical experiments with solitons and breather solutions.     

Explicit solution and Darboux transformation for a new discrete integrable soliton hierarchy with 4×4 Lax pairs

The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd.     

Integrable decompositions for the (2+1)-dimensional Gardner equation

In this paper, with the computerized symbolic computation, the nonlinearization technique of Lax pairs is applied to find the integrable decompositions for the (2+1)-dimensional Gardner [(2+1)-DG] equation. First, the mono-nonlinearization leads a single Lax pair of the (2+1)-DG equation to a generalized Burgers hierarchy which is linearizable via the Hopf–Cole transformation. Second, by the binary nonlinearization of two symmetry Lax pairs, the (2+1)-DG equation is decomposed into the generalized coupled mixed derivative nonlinear Schrödinger (CMDNLS) system and its third-order extension. Furthermore, the Lax representation and Darboux transformation for the CMDNLS and third-order CMDNLS systems are constructed. Based on the two integrable decompositions, the resonant N-shock-wave solution and an upside-down bell-shaped solitary-wave solution are obtained and the relevant propagation characteristics are discussed through the graphical analysis.     

Integrability of a Multicomponent Coupled Dispersionless Integrable System

Theoretical and Mathematical Physics - We present a multicomponent coupled dispersionless integrable system and show that it is integrable in the sense of the existence of a Lax pair representation...     

广义耦合KdV孤子方程的达布变换及其偶孤子解

根据广义耦合KdV孤子方程的Lax对, 借助谱问题的规范变换, 一个包含多参数的达布变换被构造出来. 利用达布变换来产生广义耦合KdV孤子方程的偶孤子解, 并且用行列式的形式来表达广义耦合KdV孤子方程的偶孤子解. 作为应用, 广义耦合KdV孤子方程的偶孤子解的前两个例子被给出.     

Darboux transformation and explicit solutions for the Satsuma-Hirota coupled equation

A Darboux transformation for the Satsuma-Hirota coupled equation is obtained with the help of the gauge transformation between the Lax pairs. As an application of the Darboux transformation, we give some new explicit solutions, including rational solutions, soliton solutions and periodic solutions and others, of the Satsuma-Hirota coupled equation.     

The multi-soliton solutions of the CH-γ equation

This paper solves the integrable CH-γ equation for analytical multiple soliton solutions with the Darboux transformation method. Some properties of the soliton solutions are different from the CH equation. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10401022) and the Research Grants Council of Hong Kong     

New Darboux transformation for Hirota–Satsuma coupled KdV system

A new Darboux transformation (DT) is presented for the Hirota–Satsuma coupled KdV system. It is shown that this DT can be constructed by means of two methods: Painlevé analysis and reduction of a binary DT. By iteration of the DT, the Grammian type solutions are found for the coupled KdV system.     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras I

We study coupled systems of nonlinear wave equations from the point of view of their formal Darboux integrability. By making use of Vessiot's geometric theory of differential equations, it is possible to associate to each system of nonlinear wave equations a module of vector fields on the second-order jet bundle — the Vessiot distribution. By imposing certain conditions of the structure of the Vessiot distributions, we identify the so-called separable Vessiot distributions. By expressing the separable Vessiot distributions in a basis of singular vector fields, we show that there are, at most, 27 equivalence classes of such distributions. Of these, 14 classes are associated with Darboux integrable nonlinear systems. We take one of these Darboux integrable classes and show that it is in correspondence with the class of six-dimensional simply transitive Lie algebras. Finally, this later result is used to reduce the problem of constructing exact general solutions of the nonlinear wave equations understudy to the integration of Lie systems. These systems were first discovered by Sophus Lie as the most general class of ordinary differential equations which admit nonlinear superposition principles.     

广义耦合KdV孤子方程的达布变换及其奇孤子解

借助谱问题的规范变换, 给出广义耦合KdV孤子方程的达布变换,利用达布变换来产生广义耦合KdV孤子方程的奇孤子解,并且用行列式的形式来表达广义耦合KdV孤子方程的奇孤子解．作为应用,广义耦合KdV孤子方程奇孤子解的前两个例子被给出．     

Complexiton solutions of the Korteweg–de Vries equation with self-consistent sources

Complexiton solutions to the Korteweg–de Vires equation with self-consistent sources are presented. The basic technique adopted is the Darboux transformation. The resulting solutions provide evidence that soliton equations with self-consistent sources can have complexiton solutions, in addition to soliton, positon and negaton solutions. This also implies that soliton equations with self-consistent sources possess some kind of analytical characteristics that linear differential equations possess and brings ideas toward classification of exact explicit solutions of nonlinear integrable differential equations.     

Bilinearization and Soliton Solutions of a Supersymmetric Multicomponent Coupled Dispersionless Integrable System

Theoretical and Mathematical Physics - We present a supersymmetric generalization of a multicomponent coupled dispersionless (SUSY-MCD) integrable system. Expanding the superfields of the SUSY-MCD...