极端水波的数学问题
Mathematical Aspects of Extreme Water Waves
摘要：
本论文主要研究在水动力实验室的波浪水池中产生确定性畸形波的一些理论问题。我们采用空间非线性薛定谔方程作为数学模型来描述波包包络在向下游传播时的变形。我们广泛研究了一族精确解描述调制不稳定性，被称为Akhmediev-Eleonski \u的Kulagin呼吸。与Kuznetsov-Ma呼吸子和Peregrine解一起，它们属于一类称为“非零背景孤子”的解。我们提出的动力学使用位移相位振幅表示的变分制剂。从相应的孤子物理波场出发，我们观察到振幅消失、相位奇异性和波前错位等线性现象同时出现，而这些现象的必要条件是Chu-Mei商的无界性。在荷兰海事研究所的高速波盆地进行的实验结果显示出显着的定性协议与预测的理论模型，包括振幅增加，相位奇异性，和保存的波包频率。进一步的研究表明，演化方程的一些局限性，因为它保持了对称的波信号和波的频谱在整个下游传播，而不是表现出不对称的波结构和频率下移现象。
英文摘要：
This thesis deals with some theoretical aspects of deterministic freak wave generation in the wave basin of a hydrodynamic laboratory. We adopt the spatial nonlinear Schrödinger equation as a mathematical model to describe the deformation of the wave packet envelope while propagating downstream. We study extensively a family of exact solutions describing modulational instability, known as the Akhmediev-Eleonski\uı-Kulagin breather. Together with the Kuznetsov-Ma breather and Peregrine solution, they belong to a class of solutions called ''solitons on a non-vanishing background''. We present the dynamics using the variational formulation of the displaced phase-amplitude representation. From the corresponding physical wave field of the soliton, we observe that the linear phenomena of vanishing amplitude, phase singularity, wavefront dislocation occur simultaneously and a necessary condition for these is the unboundedness of the Chu-Mei quotient. The experimental results conducted at the high-speed wave basin of Maritime Research Institute Netherlands show a remarkable qualitative agreement with the predicted theoretical model, including an amplitude increase, phase singularity, and the preservation of wave packet frequencies. A further examination suggests some limitations of the evolution equation since it maintains a symmetry in the wave signal and wave spectrum throughout downstream propagation instead of exhibiting an asymmetric wave structure and the frequency downshift phenomenon.
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