Do envelope solitons radiate

Abstract

In dispersive wave systems, when leading-order nonlinear and dispersive effects, are taken into account the envelope of a small-amplitude narrow-band wave pulse is known to satisfy the nonlinear Schrodinger (NLS) equation which, under certain conditions, admits envelope-soliton solutions. These solitons describe locally confined wave groups with envelopes of permanent form and find applications in various physical contexts. Here, is addressed the question of whether NLS envelope solitons survive when higher-order effects are taken into account. Based on a kinematic argument first, it is suggested that oscillatory tails are inevitably emitted, and this claim is further supported by numerical computations by use of a fifth-order Korteweg-deVries equation as a simple example. The radiation of tails is caused by a resonance mechanism that lies beyond all orders of the usual multiple-scale expansion leading to the NLS equation, and a procedure for calculating these tails by use of exponential asymptotics is outlined. Despite having exponentially small amplitude in the asymptotic sense, the radiated tails can be significant when pulses of relatively short duration are considered.