Discrete rogue waves and blow-up from solitons of a nonisospectral semi-discrete nonlinear Schrödinger equation

Two New Integrable Lattice Hierarchies Associated With A Discrete Schrodinger Nonisospectral Problem and Their Infinitely Many Conservation Laws

In this article, by means of using discrete zero curvature representation and constructing opportune time evolution problems, two new discrete integrable lattice hierarchies with n-dependent coefficients are proposed, which related to a new discrete Schrödinger nonisospectral operator equation. The relation of the two new lattice hierarchies with the Volterra hierarchy is discussed. It has been shown that one lattice hierarchy is equivalent to the positive Volterra hierarchy with n-dependent coefficients and another lattice hierarchy with isospectral problem is equivalent to the negative Volte…

与一类新的高维列向量环代数相关的高维等谱-非等谱可积族的生成

Generation of higher-dimensional isospectral-nonisospectral integrable hierarchies associated with a new class of higher-dimensional column-vector loop algebras

本文提出基于新构造的高维列向量环代数生成高维等谱-非等谱可积族的方法，并推导出可约化为多个经典方程（如导数非线性薛定谔方程、热方程等）的广义与扩展薛定谔族及ZN族，同时用二次型迹恒等式讨论其哈密顿结构。

Periodic energy switching of bright solitons in mixed coupled nonlinear Schr{ö}dinger equations with linear self and cross coupling terms

The bright soliton solutions of the mixed 2-coupled nonlinear Schr{ö}dinger (CNLS) equations with linear self and cross coupling terms have been obtained by identifying a transformation that transforms the corresponding equation to the integrable mixed 2-CNLS equations. The study on the collision dynamics of bright solitons shows that there exists periodic energy switching, due to the coupling terms. This periodic energy switching can be controlled by the new type of shape changing collisions of bright solitons arising in mixed 2-CNLS system, characterized by intensity redistribution, amplitud…

Shape changing (intensity redistribution) collisions of solitons in mixed coupled nonlinear Schr{ö}dinger equations

A novel kind of shape changing (intensity redistribution) collision with potential application to signal amplification is identified in the integrable $N$-coupled nonlinear Schr{ö}dinger (CNLS) equations with mixed signs of focusing and defocusing type nonlinearity coefficients. The corresponding soliton solutions for N=2 case are obtained by using Hirota's bilinearization method. The distinguishing feature of the mixed sign CNLS equations is that the soliton solutions can both be singular and regular. Although the general soliton solution admits singularities we present parametric conditions …

Deep-water waves: On the nonlinear Schr{ö}dinger equation and its solutions

We present a brief discussion on the nonlinear Schr{ö}dinger equation for modeling the propagation of the deep-water wavetrains and a discussion on its doubly-localized breather solutions that can be connected to the sudden formation of extreme waves, also known as rogue waves or freak waves.

Period preserving nonisospectral flows and the moduli space of periodic solutions of soliton equations

Flows on the moduli space of the algebraic Riemann surfaces, preserving the periods of the corresponding solutions of the soliton equations are studied. We show that these flows are gradient with respect to some indefinite symmetric flat metric arising in the Hamiltonian theory of the Whitham equations. The functions generating these flows are conserved quantities for all the equations simultaneously. We show that for 1+1 systems these flows can be imbedded in a larger system of ordinary nonlinear differential equations with a rational right-hand side. Finally these flows are used to give a co…