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Solitary waves for the Whitham equation on the whole line Stefanov, Atanas

Description

We consider the Whitham equation on the whole line. Due to the smoothing nature of the linear operator, the question for existence of traveling wave solutions has been open till recently. In 2012, Ehrnstroem-Groves-Wahlen (EGW) have constructed such waves, but only for values of $c$ slightly bigger than one, even though the admissible range of wave speeds is $c \in (1,2)$. The approach in EGW consists of a tour de force calculus of variations, supplemented by a bifurcation argument from the small KdV waves. Note that the EGW waves are of small amplitude. In this work (joint with M. Ehrnstroem), we construct a one parameter family of such waves, with wave speeds $c \in (1, c_0)$ for some limiting value $c_0$, not necessarily close to $1$. We conjecture that the wave with speed $c_0$ is the maximal amplitude wave (i.e. the highest wave, with amplitude $c/2$) and there are no waves with wave speeds $c \in (c_0,2)$. However, we still find some interesting objects in the interval $(c_0,2)$. The argument uses calculus of variation construction, very different than the one employed by EGW. It is based on constraints on appropriately selected Orlicz spaces. Finally, all our traveling waves are shown to be bell-shaped, confirming the available numerical evidence. I will also speculate a bit about their stability as they are constructed as ground states of the appropriate constrained maximization problems.

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