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Isospectral kinetic equation for solitons: integrability, exact solutions and physical applications

孤子的等谱动力学方程：可积性、精确解和物理应用

基本信息

批准号：
EP/E040160/1
负责人：
Gennady El
金额：
$ 2.05万
依托单位：
Loughborough University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FE040160%2F1
关键词：
Isospectral kinetic equation solitons integrability

项目摘要

The idea of introducing statistical description into soliton theory has two well established physical premises: a) natural wave phenomena are often so complex that they must be described statistically; b) integrable wave equations capture important qualitative and quantitative features of nonlinear wave propagation in dispersive media. By bringing together these two premises, one arrives at the challenging problem of an adequate mathematical description of the behaviour of disordered soliton systems with large number of degrees of freedom. Although the first works in this direction had been published in early 1970s, only recently a substantial progress has been achieved by considering a special thermodynamic type limit for the nonlinear modulation equations associated with the (integrable) dynamics preserving spectral parameters of the interacting waves. In the thermodynamic limit, the nonlinear interacting modes transform into randomly distributed localised states (solitons) and the modulation system assumes the form of a nonlinear kinetic equation for a soliton gas. This new kinetic equation has nontrivial mathematical structure (which is drastically different from Bolzmann's kinetic equation) and a potential for various physical applications. Both are virtually unexplored. This project is set to establish main mathematical properties of the isospectral kinetic equation for solitons and to explore its possible applications to fluid dynamics and nonlinear optics. The connections with some actively studied mathematical objects such as hydrodynamic chains and two-dimensional dispersionless hierarchies will be studied. Exact solutions will be constructed and their physical implications will be investigated. The results of the project will be of considerable interest for the nonlinear wave community in general as the isospectral kinetic equation for solitons represents a universal mathematical model applicable in different physical contexts.

将统计描述引入孤子理论的想法有两个公认的物理前提：a)自然波动现象非常复杂，必须用统计方法来描述；b)可积波动方程捕捉到了非线性波在色散介质中传播的重要定性和定量特征。通过将这两个前提结合在一起，人们就达到了一个具有挑战性的问题，即对具有大量自由度的无序孤子系统的行为进行适当的数学描述。虽然这方面的第一批工作早在20世纪70年代初就已经发表，但直到最近，通过考虑与保持相互作用波的光谱参数的(可积)动力学相关的非线性调制方程的特殊热力学类型的极限，才取得了实质性的进展。在热力学极限下，非线性相互作用模式转变为随机分布的局域态(孤子)，调制系统呈现出孤子气体的非线性动力学方程的形式。这种新的动力学方程具有非平凡的数学结构(这与玻尔兹曼的动力学方程有很大的不同)，并具有各种物理应用的潜力。这两个几乎都是未被开发的。本项目旨在建立孤子等谱动力学方程的主要数学性质，并探索其在流体力学和非线性光学中的可能应用。我们将研究它们与一些活跃的数学对象的联系，如流体动力链和二维无色散族。我们将构建精确的解，并研究它们的物理含义。由于孤子等谱动力学方程代表了一个适用于不同物理环境的通用数学模型，因此该项目的结果将对一般的非线性波群产生相当大的影响。