Bifurcation of Weakly Dispersive Partial Differential Equations

Abstract

In this thesis we explore the use of local bifurcation theory toshow existence of small-amplitude traveling wave solutions to nonlinear dispersive partial differential equations that in a sense are generalizationsof the Korteweg–de Vries and Whitham equations of hydrodynamics. Of note is the equation given by ∂_tu+L∂_xu+∂_x(u)^(p+1)= 0,whose traveling wave solutions are found to be small perturbations in thedirection of cos(ξ_0x) in the Hölder space C^{0,\alpha}(R) viewed as a bifurcation space for the problem. One of the main goals of the thesis was to provide a coherent exposition to the material needed to understand everything discussed.