Nonlinear systems: algebraic structures and integrability

非线性系统：代数结构和可积性

基本信息

批准号：
EP/X018784/1
负责人：
Simon Malham
金额：
$ 6.33万
依托单位：
Heriot-Watt University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2023
资助国家：
英国
起止时间：
2023 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FX018784%2F1
关键词：
Nonlinear systems algebraic structures integrability

项目摘要

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.

提出的研究涉及求解非线性偏微分方程的方法。这些方程用于模拟自然/物理世界中的许多复杂现象。此类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于这些企业来说至关重要。然而，找到非线性方程的解是出了名的困难。一种现在著名的经典方法，用于寻找某些类型的非线性方程的精确解，称为逆散射变换，自六十年代以来就已出现。该方法本质上将求解过程分解为求解两个线性方程的组合：一个称为 Gel'fand-Levitan-Marchenko 方程的线性积分方程和相关非线性偏微分方程的线性化版本。这种方法生成了著名的孤子解，即模拟浅水波的 Korteweg-de Vries 方程和模拟光纤中脉冲传播的非线性薛定谔方程。这样的方程被认为是“可积的”。提议者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线索。首先，该过程的简化版本很容易生成大类非局部非线性偏微分方程的解，特别包括特定类别的凝固方程。这些方程模拟了簇的形成，例如血液凝固或聚合或纳米颗粒表面沉积。其次，通过抽象求解过程，提出者展示了 Korteweg-de Vries 和非线性薛定谔方程的可积性如何等效于在相关组合代数中建立多项式展开式的存在性。本文提出的研究旨在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血方程的解，并将其用于所指出的一些应用中； (ii) 扩展抽象过程以包括所有主要的已知经典可积方程，并通过开始对适合所开发的抽象框架的可能系统进行分类，使用它来建立新的可积方程。项目（i）和（ii）代表了全新的科学。提议者还将开始寻求在这些过程和基于随机过程的非线性方程的解表示之间建立联系。目的是根据此处确定的结果提交相应的更大的拨款。