Research Projects

Following advances in two-dimensional water waves, we began to study three-dimensional effects in propagating water waves, where the dependence upon the second horizontal dimension is force by boundary or initial conditions. We study the effects of nonlinearity on the interaction of two equal but non-parallel wave trains or the reflection of an obliquely incident wave train (studied extensively by Sajjadi and Ross).

A special interest here is twofold: (i) the extension of the classical Benjamin-Fier instability stability a fully nonlinear 3D waves, and (ii) using Zakharov equations, valid for weak nonlinearity and slow modulations by solving the cubic nonlinear Schrodinger equation. This equation can be used for a study of stability and bifurcation, and the two-dimensional form has attracted a great deal of attention because it can be solved exactly by inverse scattering and other techniques.