Transverse instability of concentric soliton waves

Abstract

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 Schr\"{o}dinger 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.