Nonlinear systems: algebraic structures and integrability

本研究提出了求解非线性偏微分方程组的方法。这样的方程被用来模拟自然/物理世界中的许多复杂现象。这类模型的目标是做出预测，找到最优解决方案，甚至可能控制结果。能够高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，找到这样的非线性方程的解是出了名的困难。自六十年代以来，一种现在著名的寻找某些类型的非线性方程的精确解的经典方法，被称为逆散射变换，已经出现了。该方法实质上将求解过程分解为求解两个线性方程的组合：称为Gel‘fand-Levitan-Marchenko方程的线性积分方程和有关的非线性偏微分方程的线性化版本。这种方法产生了著名的孤子解，即模拟浅水波的Korteweg-de Vries方程和模拟脉冲在光纤中传输的非线性薛定谔方程。这样的方程被认为是“可积的”。Proposer最近的研究从两个方面为这个经典的求解过程提供了一些新的曙光。首先，简化版本的程序很容易产生大类非局部非线性偏微分方程解，特别是包括特定类型的凝血方程。这样的方程模拟了凝血或聚合或纳米颗粒表面沉积等过程中的团簇形成。其次，通过抽象求解过程，证明了Korteweg-de Vries方程和非线性薛定谔方程的可积性等价于建立了组合代数中多项式展开式的存在性。本文提出的研究试图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凝聚方程的一般类别的解，并将其用于所示的一些应用；以及(Ii)扩展抽象程序以包括所有主要的已知经典可积方程，并通过开始对符合所开发的抽象框架的可能系统进行分类来建立新的可积方程。项目(I)和(Ii)代表了全新的科学。提出者还将开始寻求在这些程序和基于随机过程的这种非线性方程的解表示之间建立联系。其目的是根据本文确定的结果提交相应的更大数额的赠款。

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.