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

Abstract

We give an exhaustive characterization of singular weak solutions for some singular ordinary differential equations. Our motivation stems from the fact that in the context of hydrodynamics several prominent equations are reducible to an equation of this form upon passing to a moving frame. We construct peaked and cusped waves, fronts with finite-time decay and compact solitary waves. We prove that one cannot obtain peaked and compactly supported traveling waves for the same equation. In particular, a peaked traveling wave cannot have compact\ud support and vice versa. To exemplify the approach we apply our results to the Camassa-Holm equation and the equation for surface waves of moderate amplitude, and show how the different types of singular solutions\ud can be obtained varying the energy level of the corresponding planar Hamiltonian systems.Peer ReviewedPostprint (published version

Traveling wave solutions of a highly nonlinear shallow water equation

“…Furthermore, we obtain negative smooth as well as anti-peaked and anti-cusped solitary waves with compact support, that is, the wave profile attains a constant value outside a finite interval. For comparison notice that it was shown to be impossible for moderate amplitude models to have peaks with compact support in [21]. Finally let us assert that we also recover all known smooth and singular traveling wave solutions of CH type equations.…”

“…[31]. Moreover, equation (1) admits peaked and cusped solutions with compact support, which was shown to be impossible for CH type equations in [21]. As we will see, the existence of such solutions requires the presence of third order terms exhibiting nonlinearities of at least cubic order in the evolution equation.…”

“…Shallow water models for waves of moderate amplitude, i.e. ε = O(δ), such as the CH [4], the corresponding equation for free surface waves [8,10,26] as well as the DP [15], capture stronger nonlinear effects and admit also non-smooth solutions containing so-called peaks and cusps, see for instance [21,31,32]. For the present equation we discover entirely new kinds of traveling wave solutions, which are not governed by equations for moderate amplitude waves.…”

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].…”

“…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].…”