On nonlinear wave envelopes of permanent form near a caustic

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Abstract
In the vicinity of a caustic of a dispersive wave system, where the group velocity is stationary and hence dispersive effects are relatively weak, the nonlinear Schrödinger equation (NLS) breaks down, and the propagation of the envelope of a finiteamplitude wavepacket is governed by a modified nonlinear Schrödinger equation (MNLS). On the basis of the MNLS, a search for wave envelopes of permanent form is made near a caustic. It is shown that possible solitary wave envelopes satisfy a nonlinear eigenvalue problem. Numerical evidence is presented of symmetric, double-hump solitary-wave solutions. Also, a variety of periodic envelopes are computed. These findings are discussed in connection with previous analytical and numerical work.