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

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)自然波现象往往非常复杂，必须用统计方法来描述；B)可积波动方程捕捉了非线性波在色散介质中传播的重要定性和定量特征。将这两个前提结合在一起，就可以得到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充分的数学描述。虽然在这个方向上的第一个作品在20世纪70年代早期就已经发表，但直到最近才取得了实质性的进展，即考虑了与（可积）动力学相关的非线性调制方程的特殊热力学类型极限，以保持相互作用波的光谱参数。在热力学极限下，非线性相互作用模式转化为随机分布的局域态（孤子），调制系统呈现为孤子气体的非线性动力学方程形式。这个新的动力学方程具有非平凡的数学结构（与玻尔兹曼的动力学方程截然不同）和各种物理应用的潜力。这两个地方几乎都没有被探索过。本项目旨在建立孤子等谱动力学方程的主要数学性质，并探索其在流体动力学和非线性光学中的可能应用。与一些正在研究的数学对象，如水动力链和二维无色散层次的联系将被研究。我们将构建精确的解，并研究它们的物理含义。由于孤子的等谱动力学方程代表了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非线性波群落产生相当大的兴趣。

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.