Solitary Traveling Water Waves of Moderate Amplitude

Abstract

We prove the existence of solitary traveling wave solutions for an equation describing the evolution of the free surface for waves of moderate amplitude in the shallow water regime. This nonlinear third-order partial differential equation arises as an approximation of the Euler equations, modeling the unidirectional propagation of surface water waves. We give a description of the solitary wave profiles by performing a phase plane analysis and study some qualitative features of the solutions.

Non-uniform continuity of the flow map for an evolution equation modeling shallow water waves of moderate amplitude

“…[2,6], it is not apparent how to control the solutions of (1) globally, due to the fact that this equation involves higher order nonlinearities in u and its derivatives than the CH equation. On the other hand, if one passes to a moving frame, it can be shown that there exist solitary travelling wave solutions decaying at infinity [14]. Their orbital stability has been recently studied in [9] using an approach proposed by Grillakis, Shatah and Strauss [15], which takes advantage of the Hamiltonian structure of (1).…”

Singular solutions for a class of traveling wave equations arising in hydrodynamics

“…which was first derived by Johnson [12], whose considerations were extended by Constantin and Lannes [4]. We refer to [11] for a first study of smooth solitary waves and to [10] for a more extensive characterization of TWS of equation (15). We introduce the traveling wave Ansatz u(x, t) = u(x − c t) and integrate once to obtain…”

Symmetric waves are traveling waves for a shallow water equation modeling surface waves of moderate amplitude

“…For a wellposedness result in the context of Besov spaces we refer the reader to [25], whereas a result on low regularity solutions with 1 < s ≤ 3/2 may be found in [22], and for global conservative solutions and continuation of solutions beyond wave breaking we point out [29]. The equation admits smooth as well as cusped and peaked solitary and periodic traveling wave solutions, and also solitary traveling waves with compact support [14,15,16]. Furthermore, the smooth solitary traveling wave solutions are known to be orbitally stable [10].…”