截断和弯曲重力波孤子的演化：马赫膨胀问题
Evolution of truncated and bent gravity wave solitons: the Mach expansion problem
摘要：
利用调制理论分析了Kadomtsev-Petviashvili（KPII）方程中内波和表面波重力波初始截断和弯曲线孤子的动力学行为。与以前的研究相比，斜相互作用的孤子，发展从锐角入射角，这项工作的重点是两个或三个部分线孤子，传播远离彼此的钝角入射的初值问题。尽管反向传播，显着的剩余孤子相互作用观察到新的物理后果。截断线孤子的初值问题描述了从宽通道中出现的准一维孤子，它与斜孤子的相互作用有关。弱相互作用和强相互作用的发展的解析描述得到的调制方程的局部孤子振幅和斜率的相互作用的简单波解。在弱相互作用的情况下，截断大钝角孤子的长时间演化表现出衰减的抛物线型波轮廓，其焦距随时间的增加逐渐趋近于圆柱形Korteweg-de Vries孤子.相反，强相互作用的情况下，稍微钝相互作用孤子演化成一个稳定的，一维线孤子的振幅减少的量成比例的入射斜率。这种强相互作用被认为是具有扩张角的孤子的“马赫膨胀”，与具有压缩角的孤子的众所周知的马赫反射形成对比。有趣的是，马赫膨胀和反射的临界角是相同的。KPII方程的数值模拟定量支持分析结果。
英文摘要：
The dynamics of initially truncated and bent line solitons for the Kadomtsev-Petviashvili (KPII) equation modelling internal and surface gravity waves are analysed using modulation theory. In contrast to previous studies on obliquely interacting solitons that develop from acute incidence angles, this work focuses on initial value problems for the obtuse incidence of two or three partial line solitons, which propagate away from one another. Despite counterpropagation, significant residual soliton interactions are observed with novel physical consequences. The initial value problem for a truncated line soliton-describing the emergence of a quasi-one-dimensional soliton from a wide channel-is shown to be related to the interaction of oblique solitons. Analytical descriptions for the development of weak and strong interactions are obtained in terms of interacting simple wave solutions of modulation equations for the local soliton amplitude and slope. In the weak interaction case, the long-time evolution of truncated and large obtuse angle solitons exhibits a decaying, parabolic wave profile with temporally increasing focal length that asymptotes to a cylindrical Korteweg-de Vries soliton. In contrast, the strong interaction case of slightly obtuse interacting solitons evolves into a steady, one-dimensional line soliton with amplitude reduced by an amount proportional to the incidence slope. This strong interaction is identified with the "Mach expansion" of a soliton with an expansive corner, contrasting with the well-known Mach reflection of a soliton with a compressive corner. Interestingly, the critical angles for Mach expansion and reflection are the same. Numerical simulations of the KPII equation quantitatively support the analytical findings.
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