Nonlinear systems: algebraic structures and integrability

提出的研究涉及非线性偏微分方程的求解方法。这些方程被用来模拟自然/物理世界中的许多复杂现象。这些模型的目标是进行预测，找到最佳解决方案，甚至控制结果。能够有效地找到准确或足够准确的解决方案对这些企业至关重要。然而，找到非线性方程的解是非常困难的。一个著名的经典方法，用于寻找某些类这样的非线性方程的精确解，称为逆散射变换，自六十年代以来一直存在。该方法基本上打破了解决方案的过程中解决两个线性方程的组合：一个线性积分方程称为凝胶'fand-Levitan-Marchenko方程和线性化版本的非线性偏微分方程有关。这种方法产生著名的孤子解决方案，Korteweg-de弗里斯方程模拟浅水波，和非线性薛定谔方程模拟脉冲在光纤中的传播。这类方程被称为“可积的”。提出者最近的研究从两个方面给这一经典的求解过程带来了新的启示。首先，一个简化版本的程序很容易产生的解决方案，大类的非局部非线性偏微分方程，特别是包括特定类别的凝固方程。这样的方程模拟簇形成，例如在血液凝固或聚合或纳米颗粒表面沉积中。其次，通过抽象的解决方案的过程中，提议者已经表明，可积的Korteweg-de弗里斯和非线性薛定谔方程是等价于建立相关的组合代数中的多项式展开的存在性。本文提出的研究试图沿着沿着这两个方向扩展这些结果，以：（i）演示如何使用简化程序的变体来确定一般类别的凝结方程的解，并在所示的一些应用中使用它;（ii）将抽象过程扩展到包括所有已知的主要经典可积方程，以及使用它来建立新的可积方程，开始分类可能的系统，适合在抽象的框架内发展。项目（i）和（ii）代表了全新的科学。提议者也将开始寻找建立这些程序之间的连接和解决方案表示为这样的非线性方程的基础上随机过程。其目的是根据本文确定的结果提交相应的更大赠款。

The research proposed concerns methods for solving nonlinear partial differential equations. Such equations are used to model many sophisticated phenomena in the natural/physical world. The goals of such models are to make predictions, find optimal solutions or maybe even to control outcomes. Being able to find exact or sufficiently accurate solutions efficiently is crucial to these enterprises. However, finding such solutions to nonlinear equations is notoriously difficult. A now famous classical method for finding exact solutions to some classes of such nonlinear equations, known as the Inverse Scattering Transform, has been around since the sixties. The method essentially breaks the solution process into solving a combination of two linear equations: a linear integral equation known as the Gel'fand-Levitan-Marchenko equation and a linearised version of the nonlinear partial differential equation concerned. This approach generates the famous soliton solutions, of the Korteweg-de Vries equation modelling shallow water waves, and of the nonlinear Schrodinger equation modelling pulse propagation in optical fibres. Such equations are said to be "integrable".The proposer's recent research has shone some new light on this classical solution procedure, in two ways. First, that a simplified version of the procedure readily generates solutions to large classes of nonlocal nonlinear partial differential equations, including in particular, specific classes of coagulation equations. Such equations model cluster formation such as in blood clotting or polymerisation or nanoparticle surface deposition. Second, by abstracting the solution procedure, the proposer has shown how the integrability of the Korteweg-de Vries and nonlinear Schrodinger equations is equivalent to establishing the existence of polynomial expansions in an associated combinatorial algebra. The research proposed herein seeks to extend these results along these two directions, to: (i) Demonstrate how a variation on the simplified procedure can be used to determine solutions to general classes of coagulation equations and use this in some of the applications indicated; and (ii) Extend the abstract procedure to include all the main known classical integrable equations, as well as use it to establish new integrable equations by starting to classify the possible systems that fit within the abstract framework developed. Projects (i) and (ii) represent completely new science. The proposer will also begin to look to establish connections between these procedures and solution representations for such nonlinear equations based on random processes. The intention is to submit a consequential larger grant based on the results established herein.