同心孤波的横向不稳定性
Transverse instability of concentric soliton waves
AI总结：
本文研究同心孤波的横向不稳定性，推导了浅水含表面张力的近同心KdV方程和深水柱坐标下的Gross-Pitaevskii方程，通过奇异特征值分析揭示表面张力与柱几何对不稳定性的关键影响。
摘要：
无论是鹅卵石撞击水面还是水下爆炸，同心表面波都不可避免地会传播。除撞击早期可能的情况外，有限振幅的同心水波产生于色散或非线性之间的平衡，从而形成孤波。尽管深水和浅水中平面孤波的稳定性已被广泛研究，但尚无针对同心孤波的类似分析。在浅水中，控制孤子形成的方程——近同心Korteweg-de Vries方程——此前已在无表面张力情况下推导出，因此我们将推导扩展到含表面张力情形。在深水中，包络方程传统上被认为是最初在笛卡尔坐标中导出的非线性薛定谔型方程。然而，通过适用于研究同心波的柱坐标中的系统推导，我们证明适当的包络方程必须修正添加一个反平方势，从而得到Gross-Pitaevskii方程。\n 我们详细研究了深水和浅水两种情形的模型性质，包括守恒律和对应于轴对称孤波的基态。后者的稳定性分析导致奇异特征值问题，这需要使用解析工具。我们确定了导致同心孤子横向不稳定的条件，揭示了它们与平面孤子的关键差异。这里特别关注的是表面张力和柱几何对横向不稳定性发生的影响。
英文摘要：
Should it be a pebble hitting water surface or an explosion taking place underwater, concentric surface waves inevitably propagate. Except for possibly early times of the impact, finite amplitude concentric water waves emerge from a balance between dispersion or nonlinearity resulting in solitary waves. While stability of plane solitary waves on deep and shallow water has been extensively studied, there are no analogous analyses for concentric solitary waves. On shallow water, the equation governing soliton formation -- the nearly concentric Korteweg-de Vries -- has been deduced before without surface tension, so we extend the derivation onto the surface tension case. On deep water, the envelope equation is traditionally thought to be the nonlinear Schrödinger type originally derived in the Cartesian coordinates. However, with a systematic derivation in cylindrical coordinates suitable for studying concentric waves we demonstrate that the appropriate envelope equation must be amended with an inverse-square potential, thus leading to a Gross-Pitaevskii equation instead. Properties of both models for deep and shallow water cases are studied in detail, including conservation laws and the base states corresponding to axisymmetric solitary waves. Stability analyses of the latter lead to singular eigenvalue problems, which dictate the use of analytical tools. We identify the conditions resulting in the transverse instability of the concentric solitons revealing crucial differences from their plane counterparts. Of particular interest here are the effects of surface tension and cylindrical geometry on the occurrence of transverse instability.
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