Waves in deep water based on the nonlinear Schrödinger equation with variable coefficients

Abstract

It has been shown that progression of waves in deep water is described by the nonlinear Schrödinger equation with time-dependent diffraction and nonlinearity coefficients. Investigation of the solutions is done here in the two cases when the coefficients are proportional or otherwise. In the first case, it is shown that the water waves are traveling at time-dependent speed and are periodic waves, which are coupled to solitons or elliptic waves seen in the noninertial frames. In the inertial frames wave modulation instability is visualized. In the second case, and when the diffraction coefficient dominates the nonlinearity, water waves collapse with unbounded amplitude at finite time. Exact solutions are found here by using the extended unified method together, while presenting a new algorithm for treating nonlinear coupled partial differential equations.