Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

Abstract

We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces ${\mathrm{H}}^s$, $s>0$, to a class of nonlinear, dispersive evolution equations of the form \begin{equation*} u_t + \left(Lu+ n(u)\right)_x = 0, \end{equation*} where the dispersion $L$ is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse $K$ and the nonlinearity $n$ is inhomogeneous, locally Lipschitz and of superlinear growth at the origin. This generalises earlier work by Ehrnström, Groves & Wahlén on a class of equations which includes Whitham's model equation for surface gravity water waves featuring the exact linear dispersion relation. Tools involve constrained variational methods, Lions' concentration-compactness principle, a strong fractional chain rule for composition operators of low relative regularity, and a cut-off argument for $n$ which enables us to go below the typical $s>\frac{1}{2}$ regime. We also demonstrate that these solutions are either waves of elevation or waves of depression when $K$ is nonnegative, and provide a nonexistence result when $n$ is too strong.

arXiv commentRevised version

References in corpus (6)

• On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type
• On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation
• Symmetry and decay of traveling wave solutions to the Whitham equation
• Convexity of Whitham's highest cusped wave
• Non-uniform dependence on initial data for the Whitham equation
• Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

Cited by in corpus (1)

• Enhanced existence time of solutions to evolution equations of Whitham type