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Modulating pulse solutions to the water-wave problem

水波问题的调制脉冲解决方案

基本信息

批准号：
229717591
负责人：
Professor Dr. Mark Groves
金额：
--
依托单位：
Matematikcentrum
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2012
资助国家：
德国
起止时间：
2011-12-31 至 2016-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/229717591?language=en
关键词：
Modulating pulse solutions water wave

项目摘要

The classical water wave problem concerns the irrotational flow of a perfect fluid of unit density subject to the forces of gravity and surface tension. A modulating pulse is a pulse-like envelope steadily advancing in the laboratory frame and modulating an underlying periodic wavetrain. The existence of modulating pulse solutions to the water-wave equations has been predicted on the basis of simplified model equations and numerics, and there has been speculation concerning their importance in the theory of freak waves. Our objective is to develop a rigorous existence theory for them. More specifically, we wish to carry out a programme of research for water waves which has been successfully used to construct modulating pulse solutions to nonlinear wave equations, namely(i) formulate the problem as an evolutionary equation in which the horizontal spatial coordinate in a frame moving with the pulse plays the role of time (`spatial dynamics');(ii) confirm the presence of an invariant finite-dimensional subspace to an approximating system which contains homoclinic solutions (these are the modulating pulses predicted by simplified model equations);(iii) establish the existence of solutions to our evolutionary system which are approximated by the above homoclinic solutions over very long length scales and for all times.

经典的水波问题涉及到单位密度的理想流体在重力和表面张力作用下的无旋流动。调制脉冲是在实验室坐标系中稳定前进并调制潜在周期波列的脉冲状包络。基于简化的模型方程和数值方法，人们预测了水波方程调制脉冲解的存在性，并推测了它们在畸形波理论中的重要性。我们的目标是为他们发展一个严格的存在理论。更具体地说，我们希望进行一项水波研究计划，该计划已成功地用于构造非线性波动方程的调制脉冲解，即（i）将问题表述为演化方程，其中随脉冲移动的框架中的水平空间坐标起时间的作用（“空间动力学”）;（ii）确认存在一个不变的有限维子空间的近似系统，其中包含同宿解（这些是通过简化的模型方程预测的调制脉冲）;（iii）建立我们的演化系统的解的存在性，这些解在很长的长度尺度上和在所有时间上都由上述同宿解近似。