关于某些方程的非线性波解的研究

This project will carry out some research on the properties of solutions for nonlinear wave equations. First, applying the theories of monotone dynamical systems and geometric singular perturbation, we will study the existence of travelling wavefronts for the generalized Burgers-Huxley equation with long-range diffusion. Second, for periodic Cauchy problems, we will study the non-uniform continuity of the solution map for periodic mCH equation in Sobolev spaces. The method is based on the approximate solutions and well-posedness estimates. Finally, since the approximate solutions for non-periodic Cauchy problem are more complex than periodic problems, we will study the non-uniform continuity for non-periodic generalized DP equation by constructing high frequency part and low frequency part for approximate solutions.

本项目将对非线性波方程的解性态进行若干研究. 首先利用单调动力系统和几何奇异摄动理论研究带长期扩散效果的广义Burgers-Huxley方程等波前解的存在性, 并利用Matlab进行数值模拟. 其次对mCH方程等周期Cauchy问题, 通过构造近似解以及做实际解的适定性估计, 在Sobolev空间中研究其解对初值的非一致连续性. 最后由于非周期Cauchy问题的近似解要比周期问题复杂的多, 通过构造近似解的高频与低频两部分, 研究广义DP方程等非周期问题解对初值在Sobolev空间中的非一致连续性.

本项目的研究工作按照计划进行并顺利完成. 首先利用几何奇异摄动理论研究带长期扩散效果的广义Burgers-Huxley 方程等波前解的存在性, 并利用Matlab 进行数值模拟. 该工作论文已投稿；其次对mCH 方程等周期Cauchy 问题, 通过构造近似解以及做实际解的适定性估计, 在Sobolev 空间中研究其解对初值的非一致连续性. 该研究工作已经发表在SCI杂志NoDEA上([Nonlinear Differ. Equ. Appl. 20 (2013), 741–755]). 项目研究历时一年,发表论文一篇.