Modulation of trapped waves giving approximate two-dimensional solitions

Abstract

In a previous paper, F. Calogero and the author studied the non-linear modulation of dispersive trapped water-waves in a channel or along a beach (edge waves). They showed that the signal envelope, (which is studied with convenient time scale and length scale) obeys Nonlinear Schrödinger Equation. The result is extended here to waves trapped by the bottom geometry of a basin which is infinitely extended into all horizontal directions. One can obtain along the guided path, envelope solitons of Nonlinear Schrödinger Equation for a signal whose amplitude exponentially decreases in transverse directions. These bidimensional solitons are thus shown as asymptotic features for a large class of problems. This class is narrower than in the finite case or in the semiinfinite case because strong conditions must be set either on the linear part or on the nonlinear part of the operator in order to guarantee an exponentially decreasing behavior on both sides. In addition, the envelope solitons are in most cases unstable features, because of side-band instabilities. Nevertheless, the result suggests that the exact bidimensional solitons recently derived by Boiti et al. for special Equations are not an exceptional physical feature of two-dimensional evolutions.