具有不光滑孤子解非线性色散波方程的奇性解和全局解

项目名称： 具有不光滑孤子解非线性色散波方程的奇性解和全局解

项目编号： No.11471259

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 付英

作者单位： 西北大学

项目金额： 68万元

中文摘要： 具有尖峰孤子解的方程一直是浅水波方程研究中数学家和物理学家非常关注的，而自从Camassa-Holm方程被发现以来，具有尖峰孤子解的推广Camassa-Holm型方程层出不穷。本项目主要研究几个推广Camassa-Holm方程的爆破解和整体解，即我们构造的三个模型：推广两分量Camassa-Holm方程，μ形式的带有三次非线性项Camassa-Holm方程以及μ形式Fokas方程，它们全局强解和全局弱解的存在性，周期情形下全局守恒解和全局耗散解的存在性和唯一性，改进的爆破结果；μ-Camassa-Holm方程和μ-Degasperis-Procesi方程更新的爆破充分条件，全局守恒解和全局耗散解的存在性和唯一性。这些推广方程解的性质对于研究具有强非线性作用的浅水波提供一定的理论指导，而这些新模型孤子解的性质对研究相关的物理现象具有一定的指导作用。

中文关键词： Camassa-Holm；方程；Degasperis-Procesi；方程；尖峰孤子解；强解的爆破；整体解

英文摘要： In the study of shallow water equations, the equations with peakons have always attracted the attention of mathematicians and physicists. Since the Camassa-Holm equation was discovered, there appeared various generalized Camassa-Holm equations with peakons. In this project, we are mainly concerned with the blow-up and global solutions to several generalized Camassa-Holm equations: the existence of global strong and weak solutions, the existence and uniqueness of global conservative solutions and global dissipative solutions in periodic case and improved blow-up results of strong solutions to the generalized 2-component Camassa-Holm equation, the μ-version Camassa-Holm equation with cubic nonlinearity and the μ-version Fokas equation all of which have been constructed by us; renewed blow-up conditions of strong solutions, the existence and uniqueness of global conservative solutions and global dissipative solutions to μ-Camassa-Holm equation and μ-Degasperis-Procesi equation. These properties of solutions to the generalized Camassa-Holm equations will provide the theoretical direction for studying the shallow water waves with strong interaction. Moreover, the research with regard to the properties of peakons to these new models will play a certain role to the related physical phenomena.

英文关键词： Camassa-Holm equaation;Degasperis-Procesi equation;Peaked solitons;Blow-up of strong solutions;Global solutions