非局部孤立子方程的有理解

Soliton, rogue wave and breather as nonlinear phenomena frequently appear in optics, water wave, plasma physics, etc. The study of the dynamical for the solutions in soliton equation is a hot topic in mathematics, physics and other fields. Because of the complexity of solving partial differential equations, there is not a simple and unique method to construct the rational solutions of the nonlocal soliton equation. So, it is difficult to discuss the structure of the solutions and their dynamic properties. We use the Hirota bilinear method and KP hierarchy reduced method, to improve the existed method and provide a strict proof of this method. This make the process of solving soliton solution more concise, efficient and accurate. Using this improved method, we also try to find new mixed solutions (including soliton, breather and rational solution) and analyze their rich dynamic properties..The main research will focus on following items: 1) Improving the Hirota bilinear method and KP hierarchy reduced method. We provide Gram-type solution from the tau functions in KP hierarchy, and give a strict proof of the rational solutions. 2) According to the improved method, we construct the soliton , rogue wave(or rational solution) and mixed solutions. We discuss the dynamical characteristics of rational solutions by the analytical expression, classify the solutions. The research of this project will provide a powerful theoretical basis for the experimental research and application of the nonlinear waves in nonlocal soliton equations.

孤立子、怪波、呼吸子等非线性现象频繁出现在光学、水波、等离子体等领域，研究非线性孤立子方程的这些类型的精确解及其动力学性质是数学物理领域的重要课题。由于非线性偏微分方程求解的复杂性，还没有统一有效的非线性孤立子方程有理解的构造方法，这对解的结构以及动力学性质的讨论带来困难。本项目根据Hirota双线性方法和KP族约化方法，对已有方法改进并严格证明，使孤立子方程求解过程更简洁高效精确，并由此寻找新的混合解（由孤立子，呼吸子，有理解构成），分析其丰富的动力学性质。. 本项目的主要研究内容包括：1)改进Hirota双线性方法，综合运用KP族约化方法的基础上，给出严格的有理化过程以及证明。2)基于上述改进的求解方法，给出几类非局部孤立子方程的孤立子解，怪波解（或有理解）以及混合解。利用有理解的解析表达式讨论解的动力学性质，同时探索解的分类。本项目的研究将为实验提供有力的证据。

孤立子、怪波、呼吸子等非线性现象频繁出现在光学、水波、等离子体等领域，研究局部和非局部孤立子方程的上述类型的精确解及其动力学性质是数学物理领域的重要课题。本项目以Hirota双线性方法和KP系列约化方法为辅助研究工具，对已有方法改进，使非线性孤立子方程求解过程更简洁高效精确，进一步探讨局部和非局部孤立子方程的新混合解（由孤立子，呼吸子，有理解，lump解构成），并分析其丰富的动力学性质。.本项目着重从以下方面探讨：1）建立和完善局部和非局部Mel’nikov equation、非局部Schrodinger-Boussinesq方程等非局部孤立子方程模型的孤立子解、呼吸子解、怪波解、lump解、半有理解和混合有理解，在此基础上进一步探讨各种不同解叠加的动力学模式；2）考虑一类3+1维非线性发展方程的高阶lump解和孤立子解的裂变和聚变碰撞。3）讨论2+1维 Hirota-Satsuma-Ito 方程的高阶呼吸子解、lump解和半有理解。本项目研究成果将促进学科交叉发展，对海洋、大气、光学等复杂系统的非线性现象的研究提供理论基础和有力工具。