Nonisospectral Equation Research Articles

Nonisospectral equations from the Cauchy matrix approach

Nonisospectral equations from the Cauchy matrix approach

Generalized Tappert transformation in femtosecond nonlinear optics

We present the generalized Tappert transformation for the higher-order nonlinear Schrödinger equation model of femtosecond nonlinear optics, in which the gravitational-like potential is introduced with the aim of the effective parameterization of the Raman soliton self-scattering effect. By means of the nonlinear change of variables and nontrivial reversible gauge transformation the higher-order forced nonlinear Schrödinger equation model is turning into the “free” higher-order nonlinear Schrödinger equation without external potential, but with varying in time dispersion and nonlinearity. The proposed generalized Tappert transformations allow one to find the direct relationships between the soliton solutions of the forced nonisospectral equations and the corresponding isospectral equations without external potential, but with varying in time dispersion and nonlinearity coefficients.

Mixed spectral AKNS hierarchy from linear isospectral problem and its exact solutions

Abstract In this paper, the AKNS isospectral problem and its corresponding time evolution are generalized by embedding three coefficient functions. Starting from the generalizedAKNS isospectral problem, a mixed spectralAKNS hierarchy with variable coefficients is derived. Thanks to the selectivity of these coefficient functions, the mixed spectral AKNS hierarchy contains not only isospectral equations but also nonisospectral equations. Based on a systematic analysis of the related direct and inverse scattering problems, exact solutions of the mixed spectral AKNS hierarchy are obtained through the inverse scattering transformation. In the case of reflectionless potentials, the obtained exact solutions are reduced to n-soliton solutions. This paper shows that the AKNS spectral problem being nonisospectral is not a necessary condition to construct a nonisospectral AKNS hierarchy and that the inverse scattering transformation can be used for solving some other variable-coefficient mixed hierarchies of isospectral equations and nonisospectral equations.

Variable-coefficient nonisospectral Toda lattice hierarchy and its exact solutions

In this paper, a hierarchy of nonisospectral equations with variable coefficients is derived from the compatibility condition of Toda spectral problem and its time evolution. In order to solve the derived Toda lattice hierarchy, the inverse scattering transformation is utilized. As a result, new and more general exact solutions are obtained. It is shown that the inverse scattering transformation can be generalized to solve some other nonisospectral lattice hierarchies with variable coefficients.

Exact solutions to a mixed Toda lattice hierarchy through the inverse scattering transform

Starting from the Toda spectral problem, a mixed hierarchy of the isospectral equations and the nonisospectral equations is constructed through discrete zero curvature equations. Soliton solutions of the hierarchy are obtained by the inverse scattering transform. Meanwhile, a new kind of time evolution of scattering data for the Toda spectral problem is presented.

Symmetries for the Ablowitz-Ladik Hierarchy: Part I. Four-Potential Case

In the paper, we first investigate symmetries of isospectral and non-isospectral four-potential Ablowitz–Ladik hierarchies. We express these hierarchies in the form of un,t= LmH(0), where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. Then by means of the zero-curvature representations of the isospectral and non-isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non-isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac-Moody-Virasoro algebras. The recursion operator L is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four-potential and two-potential Ablowitz–Ladik hierarchies. The even order members in the four-potential Ablowitz–Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two-potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes L2.

THE MULTI-SOLITON SOLUTIONS FOR THE NON-ISOSPECTRAL mKdV–SINE-GORDON EQUATION

We derive multi-soliton solutions for a non-isospectral mKdV–sine-Gordon equation by means of a bilinear approach. Solutions are given in both Hirota's form and Wronskian form. This non-isospectral equation is a generic one which is related to a time-dependent spectral parameter with time evolution [Formula: see text].

Nonisospectral integrable nonlinear equations with external potentials and their GBDT solutions

Auxiliary systems for matrix nonisospectral equations, including coupled NLS with external potential and KdV with variable coefficients, were introduced. Explicit (up to the inversion of finite matrices) solutions of nonisospectral equations were constructed using the GBDT version of the Bäcklund–Darboux transformation.

A modified KdV equation with self-consistent sources in non-uniform media and soliton dynamics

Two non-isospectral modified KdV equations with self-consistent sources are derived, which correspond to the time-dependent spectral parameter λ satisfying λt = λ and λt = λ 3 , respectively. Gauge transformation between the first non-isospectral equation (corresponding to λt = λ) and its isospectral counterpart is given, from which exact solutions and conservation laws for the non-isospectral one are easily listed. Besides, solutions to the two nonisospectral modified KdV equations with self-consistent sources are derived by means of the Hirota method and the Wronskian technique, respectively. Nonisospectral dynamics and source effects, including one-soliton characteristics in non-uniform media, two-solitons scattering and special behaviours related to sources (for example, the ‘ghost’ solitons in the degenerate two-soliton case), are investigated analytically.

New symmetries for the Ablowitz–Ladik hierarchies

In the Letter we give new symmetries for the isospectral and non-isospectral Ablowitz–Ladik hierarchies by means of the zero curvature representations of evolution equations related to the Ablowitz–Ladik spectral problem. Lie algebras constructed by symmetries are further obtained. We also discuss the relations between the recursion operator and isospectral and non-isospectral flows. Our method can be generalized to other systems to construct symmetries for non-isospectral equations.

Soliton-like solutions of three non-isospectral equations

n-soliton-like solutions of three non-isospectral equations, the non-isospectral mKdV equation, the non-isospectral sine-Gordon equation and the non-isospectral nonlinear Schrodinger equation were obtained by using the Hirota method.

Mixed solutions in the wronskian form for a nonisospectral equation

N-soliton solutions in the Wronskian form for the KdV equation with loss and nonuniformity terms were obtained. New rational-like solutions and mixed solutions were further derived. All these solutions were verified by direct substitutions into bilinear e-quation.

Negatons, positons, rational-like solutions and conservation laws of the Korteweg–de Vries equation with loss and non-uniformity terms

Solitons, negatons, positons, rational-like solutions and mixed solutions of a non-isospectral equation, the Korteweg–de Vries equation with loss and non-uniformity terms, are obtained through the Wronskian technique. The non-isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using Mathematica. We also derive an infinite number of conservation laws of the equation.