Exact solutions for discrete and continuous nonlinear systems

这是一个数学的建议，在广泛的领域的可积系统，重点是精确解的非线性系统。可积系统领域始于浅水中孤立波的显着发现，称为孤子，它改变了范式和我们对非线性现象的理解。作为一个数学概念，孤子首次出现在大约50年前，当时描述浅水波的Korteweg-de弗里斯方程的解析解是通过逆散射方法明确构造的。这种方法很快被应用于许多系统，这些系统对于应用很重要，例如非线性薛定谔方程（非线性光学，调制不稳定性），正弦戈登方程（非线性光学，约瑟夫森结，蛋白质和DNA中的低频集体运动），海森堡和朗道-利夫希茨模型（磁学理论）以及许多其他模型。在过去的十年中，Kadomtsev-Petviashvili（KP）方程的孤子理论与簇代数和枚举几何的惊人联系被发现。KP方程用于模拟表面上的浅水波。其孤子解形成网状结构。儿玉和威廉姆斯用完全积极的格拉斯曼主义者来描述他们。最近，系统的偏微分和微分差分方程，我们已经开发出一种方法，用于建设的精确解的对称性的基础上的拉克斯表示和发现新的解决方案，这代表非线性波阵面传播的恒定速度。很明显，孤子的世界比人们预期的要丰富得多。配备了这种新的方法，我们已经准备好解决问题的建设，描述和可视化的基本系统的偏微分和微分差分方程的精确解。这项建议的目标是非常雄心勃勃的。该提案分为四个部分，与上述四个目标相对应。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.