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。

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.