可积系统方法在怪波及相关问题中的应用

Rogue wave phenomenon can be investigated by means of experiment-controlled nonlinear models. To construct the rogue wave solutions of the models is currently an international hot topic and issue. Based on soliton theory and with aid of the symbolic computation and numeric simulation, the project concentrates on exploring the rogue wave solutions of nonlinear models by virtue of developing the approaches of integrable systems, presenting the corresponding mechanizations and carrying out the numeric simulation. The subjects of this project mainly focus on: (1) For the Lax integrable nonlinear systems, we can seek out the general formulation of rogue wave solutions and dynamic applications by improving the Darboux transformation. (2) Using inverse-scattering transformation and Riemann-Hilbert method for the Lax integrable nonlinear systems (or in the semiclassical limit nonlinear systems), we can cook up the asymptotic formula of rogue wave solutions. Further we can analyze the properties and some related applications. (3) For the bilinear integrable nonlinear systems, we can exploit the Hirota method to create general formulation of rogue wave solutions and dynamic applications. Further we will dig up their practical applications. The results presented in this project will promote interdisciplinary research development and provide a very effective tool and theoretical basis for the study of rogue wave problems of the complicated systems in some fields, such as oceangraphy, aerography, optics, finance and so on.

怪波现象可借助于实验可控的非线性模型研究，是目前国内外研究的热点与焦点，本项目将基于孤立子理论，以符号计算、数值模拟为辅助研究工具，发展可积系统中求解方法，从三个方面发展构造非线性可积模型怪波，编制相应的推导软件包，数值模拟与现实的吻合程度：(1) 对Lax可积的非线性系统，发展新变量分离方法求解Lax对，进一步改进Darboux变换为工具寻求怪波一般表达式，并考虑其动力学特征及数值模拟；(2) 对Lax可积的非线性系统 (或在半经典极限意义下非线性系统)，以反散射变换与RH方法为工具，探索怪波的解析渐近式，并分析其渐近性质及应用；(3) 对双线性可积的非线性系统，将发展Hirota方法探索怪波的一般表达式，分析其动力学特征及数值模拟，并研究解的实际应用。本项目研究成果将促进多学科交叉发展，对海洋、大气、光学、金融等复杂系统中怪波行为的研究提供理论基础和有力工具。

怪波现象可借助于实验可控非线性模型研究，是目前国内外研究的热点与焦点，它的研究将有助于减少海难、提高天气预报准确率、降低金融风险等，本项目将基于孤立子理论，以符号计算、数值模拟为辅助研究工具，发展可积系统中求解方法，从三个方面发展构造非线性可积模型怪波解，编制相应的推导软件包：(1) 对Lax可积的非线性系统，发展新变量分离方法求解Lax对，进一步改进Darboux变换为工具寻求怪波一般表达式，并考虑其动力学特征及数值模拟(2) 对于Lax可积非线性系统，发展新的变量分离方法，进一步改进Darboux变换，探讨高阶Peregrine解，并分析其的动力学特征 (3) 对双线性可积的非线性系统，将发展Hirota方法构造怪波及lump解，分析其动力学特征及数值模拟，并研究解的实际应用。本项目研究成果将促进学科交叉发展，对海洋、大气、光学、金融等复杂系统怪波行为的研究提供理论基础和有力工具。