Solitary waves and fronts

Abstract: I will give a general introduction to the theory of solitary water waves, that is traveling waves whose surfaces converge to some asymptotic height at infinity. In many respects, the theory for solitary waves is more difficult and more subtle than that for periodic waves. Yet in other ways the problem is much simpler, and indeed many results for solitary waves are stronger than their periodic counterparts.

Beginning with the linear theory, or perhaps more accurately the lack of a linear theory, I will explain how small-amplitude waves can be rigorously constructed via a center manifold reduction. Next I will collect a series of results which together guarantee that any solitary wave, regardless of amplitude, is symmetric and decreasing about a central crest and travels at a “supercritical” speed $c>\sqrt{gd}$. Finally, I will explain how the significant obstacles to applying global bifurcation techniques can be overcome by taking advantage of the above properties together with the nonexistence of front-type solutions. This approach is surprisingly robust, and has recently been generalized to apply to front-type solutions in addition to solitary waves.

analysis of PDEsclassical analysis and ODEs

Audience: researchers in the topic

Online Northeast PDE and Analysis Seminar