Generalized Lattice Heisenberg Magnet Model and Its Quasideterminant Soliton Solutions

We consider a Darboux transformation of a generalized lattice (or semidiscrete) Heisenberg magnet (GLHM) model. We define a Darboux transformation on solutions of the Lax pair and on solutions of the spin evolution equation of the GLHM model. The solutions are expressed in terms of quasideterminants. We give a general expression for K-soliton solutions in terms of quasideterminants. Finally, we obtain one- and two-soliton solutions of the GLHM model using quasideterminant properties.     

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 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.     

Darboux Transformation and Exact Solutions for Two Dimensional A 2n−1 (2) Toda Equations

The Darboux transformation for the two dimensional A⁽²⁾_2n-1 Toda equations is constructed so that it preserves all the symmetries of the corresponding Lax pair. The expression of exact solutions of the equation is obtained by using Darboux transformation.     

一个离散晶格方程的N波Darboux变换和无穷守恒律

根据已知离散晶格方程的Lax对,构建了该方程的Ⅳ波Darboux变换和无穷守恒律,通过应用Darboux变换,得到离散晶格方程的范德蒙行列式形式的精确解,通过画图给出了该方程一类特殊的单孤子结构.     

A new lattice hierarchy: Hamiltonian structures,symplectic map and N-fold Darboux transformation

A new lattice hierarchy is constructed from a discrete matrix spectral problem. By the Tu scheme technique, the associated Hamiltonian structures and infinitely many conservation laws of this hierarchy are derived. Then a symplectic map is proposed based on the Lax pair and the adjoint Lax pair. Furthermore, the N-fold Darboux transformation and explicitly exact solutions of the first two equations in the hierarchy are investigated. Finally, the density profiles of these exact solutions are presented to illustrate the overtaking collisions of solitary waves.     

Integrable Semi-discrete Kundu–Eckhaus Equation: Darboux Transformation,Breather, Rogue Wave and Continuous Limit Theory

To get more insight into the relation between discrete model and continuous counterpart, a new integrable semi-discrete Kundu–Eckhaus equation is derived from the reduction in an extended Ablowitz–Ladik hierarchy. The integrability of the semi-discrete model is confirmed by showing the existence of Lax pair and infinite number of conservation laws. The dynamic characteristics of the breather and rational solutions have been analyzed in detail for our semi-discrete Kundu–Eckhaus equation to reveal some new interesting phenomena which was not found in continuous one. It is shown that the theory of the discrete system including Lax pair, Darboux transformation and explicit solutions systematically yields their continuous counterparts in the continuous limit.     

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.     

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.     

Darboux transformation and N-soliton solutions for a more general set of coupled integrable dispersionless system

The Darboux transformation and Lax pair of a more general set of coupled integrable dispersionless system are derived. By Darboux transformation, N-soliton solutions for the coupled integrable dispersionless system are obtained. In particular, the multi-soliton solutions are shown through some figures.     

Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation

Two hierarchies of integrable positive and negative lattice equations in connection with a new discrete isospectral problem are derived. It is shown that they correspond to positive and negative power expansions respectively of Lax operators with respect to the spectral parameter, and each equation in the resulting hierarchies is Liouville integrable. Moreover, infinitely many conservation laws of corresponding positive lattice equations are obtained in a direct way. Finally, a Darboux transformation is established with the help of gauge transformations of Lax pairs for the typical lattice soliton equations, by means of which the exact solutions are given.     

Darboux transformations for a matrix long-wave–short-wave equation and higher-order rational rogue-wave solutions

A new matrix long-wave–short-wave equation is proposed with the of help of the zero-curvature equation. Based on the gauge transformation between Lax pairs, both onefold and multifold classical Darboux transformations are constructed for the matrix long-wave–short-wave equation. Resorting to the classical Darboux transformation, a multifold generalized Darboux transformation of the matrix long-wave–short-wave equation is derived by utilizing the limit technique, from which rogue wave solutions, in particular, can be obtained by employing the generalized Darboux transformation. As applications, we obtain rogue-wave solutions of the long-wave–short-wave equation and some explicit solutions of the three-component long-wave–short-wave model, including soliton solutions, breather solutions, the first-order and higher-order rogue-wave solutions, and others by using the generalized Darboux transformation.     

色散长波方程的Darboux变换及多孤子解

根据色散长波方程的可积性,首先借助符号计算构造出该方程的Lax对,接着构建一个包含多参数的Darboux变换,通过应用Darboux变换,得到色散长波方程的2N-孤子解,最后通过图像研究了孤子解的性质,这些解和图像可能对解释色散长波方程所描述的水波现象有所帮助.     

Necessary Covariance Conditions for a One-Field Lax Pair

We study the covariance with respect to Darboux transformations of polynomial differential and difference operators with coefficients given by functions of one basic field. In the scalar (Abelian) case, the functional dependence is established by equating the Frechet differential (the first term of the Taylor series on the prolonged space) to the Darboux transform; a Lax pair for the Boussinesq equation is considered. For a pair of generalized Zakharov-Shabat problems (with differential and shift operators) with operator coefficients, we construct a set of integrable nonlinear equations together with explicit dressing formulas. Non-Abelian special functions are fixed as the fields of the covariant pairs. We introduce a difference Lax pair, a combined gauge-Darboux transformation, and solutions of the Nahm equations.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 122–132, July, 2005.     

Symbolic computation on the Darboux transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation from fiber optics

Considering the propagation of ultrashort pulse in the realistic fiber optics, a generalized variable-coefficient higher-order nonlinear Schrödinger equation is investigated in this paper. Under certain constraints, a new 3×3 Lax pair for this equation is obtained through the Ablowitz-Kaup-Newell-Segur procedure. Furthermore, with symbolic computation, the Darboux transformation and nth-iterated potential transformation formula for such a model are explicitly derived. The corresponding features of ultrashort pulse in inhomogeneous optical fibers are graphically discussed by the one- and two-soliton-like solutions.     

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．     

Darboux Transformations and Global Explicit Solutions for Nonlocal Davey–Stewartson I Equation

For the nonlocal Davey–Stewartson I equation, the Darboux transformation is considered and explicit expressions of the solutions are obtained. Like other nonlocal equations, many solutions of this equation may have singularities. However, by suitable choice of parameters in the solutions of the Lax pair, it is proved that the solutions obtained from seed solutions which are zero and an exponential function of t , respectively, by a Darboux transformation of degree n are global solutions of the nonlocal Davey–Stewartson I equation. The derived solutions are soliton solutions when the seed solution is zero, in the sense that they are bounded and have n peaks, and “dark cross soliton” solutions when the seed solution is an exponential function of t , in the sense that they are bounded and their norms change fast along some crossing straight lines.     

Bell polynomial approach to an extended Korteweg–de Vries equation

Under investigation in this paper is an extended Korteweg–de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV–SK‐type equation and KdV–Lax‐type equation. On the one hand, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV–Lax‐type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV–SK‐type equation is also presented, and the exact relation between the one periodic wave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

On a vector long wave-short wave-type model

A new vector long wave-short wave-type model is proposed by resorting to the zero-curvature equation. Based on the resulting Riccati equations related to the Lax pair and the gauge transformations between the Lax pairs, multifold Darboux transformations are constructed for the vector long wave-short wave-type model. This method is general and is suitable for constructing the Darboux transformations of other soliton equations, especially in the absence of symmetric conditions for Lax pairs. As an illustrative example of the application of the Darboux transformations, exact solutions of the two-component long wave-short wave-type model are obtained, including solitons, breathers, and rogue waves of the first, second, third, and fourth orders. All the solutions derived by the Darboux transformations involve square roots of functions, which is not observed in the investigation of other nonlinear integrable equations. This model describes new nonlinear phenomena.