代数型孤子的求解及其分析的研究

Algebraic solitons is a kind of important solutions in integrable nonlinear partial differential equations. These solutions can be used to explain some unique natural phenomena, such as rogue waves in nonlinear optical waves and water waves. Many researches sugguest that rogue waves originated from modulation instability (MI). This project will firstly study the relationship between algebraic solitons and MI in the model of derivative nonlinear Schrodinger (DNLS) equation by Darboux transformation and inverse scattering methods. After that, we will use Matveev method of solitons propagation on an arbitrary background to construct the decompositions of the high order algebraic solitons. Finally, we will develop Darboux transformation and inverse scattering methods to study algebraic solitons and the interaction of the higher order algebraic solitons of integrable nonlinear partial differential equations with PT symmetry. The researches in this program will certainly improve the effects of theories about the formation mechanism and applications of algebraic solitons, integrable systems and so on.

代数型孤子是可积非线性偏微分方程中一类重要的解。这种解可以来解释一些独特的自然现象，例如在非线性光学和水波中出现的怪波。许多研究认为怪波来源Modulation stability(MI)。本项目将首先利用Darbou变换和反散射方法以导数非线性薛定谔(DNLS)方程为模型研究代数型孤子解与MI的关系，然后我们将利用Matveev给出在任意背景下孤子的相互作用的方法来解决高阶代数型孤子解的分解。最后我们还将利用Darboux变换和反散射方法来研究PT对称的可积非线性偏微分方程中代数型孤子解的求解和高阶代数型孤子解相互作用。该项目的研究会对代数型孤子的形成机制、应用以及可积系统等理论有一定的促进作用。

代数型孤子是可积非线性偏微分方程中一类重要的解，可以用来描述某些独特的自然现象（比如非线性光学和水波中怪波）。可积非线性偏微分方程的解析解的求解及其分析是可积系统和非线性波中的重要问题。本项目主要从可积系系统的常用求解方法（Darboux方法、反散射方法和Hirota双线性方法）出发，给出导数非线性薛定谔 (DNLS) 方程等可积非线性偏微分方程的孤子型解及其退化，同时给出经典孤子解、呼吸子解和Phase解之间的关系和退化过程，并且研究高阶代数型孤子解的相互作用及其形成。这些结果有助于深刻理解可积系统理论及其复杂波的调制和怪波应用研究。