Exact solutions for discrete and continuous nonlinear systems

这是在可积系统的广泛领域中的一项数学建议，重点是非线性系统的精确解。可积系统的领域始于浅水上孤立波的显著发现，也就是我们所知的孤子，这改变了我们对一般非线性现象的范式和理解。作为一个数学概念，孤子最早出现在大约50年前，当时描述浅水波的Korteweg-de Vries方程的解析解是用逆散射方法显式构造的。这种方法很快被应用于许多具有重要应用价值的系统，如非线性薛定谔方程(非线性光学，调制不稳定性)，Sine-Gordon方程(非线性光学，超导约瑟夫森结，蛋白质和DNA中的低频集体运动)，Heisenberg和Landau-Lifshitz模型(在磁理论中)等等。在过去的十年里，Kadomtsev-Petviashvili(KP)方程的孤子理论与团簇代数和计数几何之间有着惊人的联系。用KP方程模拟水面上的浅水波。它的孤子解形成了网状结构。小玉和威廉姆斯用完全积极的格拉斯马尼亚人来描述他们。最近，对于偏微分方程组和微分差分方程组，我们发展了一种基于Lax表示的对称性来构造精确解的方法，并发现了一类新的解，它们代表了以恒速传播的非线性波前。很明显，孤子的世界比人们预期的要丰富得多。有了这套新的方法，我们已准备好处理偏微分方程组和微分差分方程组的精确解的构造、描述和可视化问题。该提案分为四个部分，与上述四项目标相对应。每个目标都可以独立实现，尽管它们是密切相关的。

This is a Mathematics proposal in the broad area of Integrable Systems with a focus on exact solutions of nonlinear systems. The area of Integrable Systems started with the remarkable discovery of solitary waves on shallow water, known as solitons, which has changed the paradigm and our understanding of nonlinear phenomena in general. As a mathematical concept, solitons first appeared about 50 years ago when analytical solutions for the Korteweg-de Vries equation, describing shallow water waves, were explicitly constructed by the inverse scattering method. This method were soon applied for many systems, which are important for applications, such as the Nonlinear Schrodinger equation (non-linear optics, modulation instability), sine-Gordon equation (non-linear optics, superconductive Josephson junctions, low-frequency collective motion in proteins and DNA), Heisenberg and Landau-Lifshitz models (in the theory of magnetism) and many others. Over the last decade surprising connections of the soliton theory for the Kadomtsev-Petviashvili (KP) equation with cluster algebras and enumerative geometry have been discovered. The KP equation is used to model shallow water waves on a surface. Its soliton solutions form web structures. Kodama and Williams described them in terms of totally positive Grassmanians. Recently, for systems of partial differential and differential-difference equations we have developed a method for construction of exact solutions based on symmetries of the Lax representations and discovered new classes of solutions, which represent nonlinear wave fronts propagating with constant velocity. It has become clear that the world of solitons is much richer than it has been anticipated. Equipped with this new methodology, we are well prepared to tackle the problem of construction, description and visualisation of exact solutions for basic systems of partial differential and differential-difference equations.The aims of this proposal are highly ambitious. The proposal is divided into four parts corresponding to the four objectives listed above. Each objective can be achieved independently, though they are closely related.