Modulating pulse solutions to the water-wave problem

经典的水波问题涉及单位密度的理想流体在重力和表面张力的作用下的无旋流动。调制脉冲是在实验室框架内稳定前进的脉冲状包络，并调制潜在的周期性波列。根据简化的模型方程和数值计算，预测了水波方程调制脉冲解的存在，并推测了它们在反常波理论中的重要性。我们的目标是为他们发展一个严格的生存理论。更具体地说，我们希望对水波开展研究，它已成功地用于构造非线性波动方程的调制脉冲解，即：(1)将问题描述为一个演化方程，其中随脉冲移动的坐标系中的水平空间坐标起着时间的作用(‘空间动力学’)；(2)证实存在一个不变的有限维子空间；(2)建立一个包含同宿解(即由简化模型方程预测的调制脉冲)的近似系统的解的存在性；(3)建立由上述同宿解在很长时间尺度上近似的演化系统解的存在性。

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.