Branching Near Plane Waves in Perturbed Dispersive Systems

Abstract

A Poincaré‐Lindstedt type technique for partial differential equations is used to study branching phenomena in perturbed dispersive systems arising in hydrodynamic stability theory. Multi‐periodic waves with two frequencies which branch from a family of neutrally stable nonlinear periodic plane waves are constructed, the second frequency as a power series expansion in ε. The branching is compared with that of the unperturbed equations described in an earlier paper for the purpose of understanding how higher order perturbation terms effect the properties of the lower order amplitude equations. We find that in general the perturbation terms alter the leading order frequency shifts, thus changing the bifurcation from pitchfork to transverse type. The method is used to study the perturbed nonlinear Schrödinger equation and the perturbed MKdV equation.