Abstract
We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces , , to a class of nonlinear, dispersive evolution equations of the formwhere the dispersion is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse and the nonlinearity is inhomogeneous, locally Lipschitz and of superlinear growth at the origin. This generalises earlier work by Ehrnström, Groves and 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 which enables us to go below the typical regime. We also demonstrate that these solutions are either waves of elevation or waves of depression when is nonnegative, and provide a nonexistence result when is too strong.
Highlights
Many model equations for one-dimensional spacial evolution of water waves [20] may be written as ut + (Lu + n(u))x = 0, (1)
U(ξ) ξ as an alternative to the Korteweg–de Vries (KdV) equation featuring the exact linear dispersion relation for unidirectional water waves influenced by gravity
We study the existence of solutions to (2) in fractional Sobolev spaces both on the real line and in the periodic setting, noting that σ = − 12, = 1 and q = 1 for the original Whitham equation
Summary
Additional analytical and numerical results for the Whitham equation include modulational instability of periodic waves [17, 29], local well-posedness in Sobolev spaces Hs, s > 32, for both solitary and periodic initial data [11, 7, 19], non-uniform continuity of the d ata-to-solution map [1], symmetry and decay of traveling waves [3], analysis of modeling properties, dynamics and identification of scaling regimes [19], and wave-channel experiments and other numerical studies [2, 5, 18, 32] These investigations have demonstrated the potential usefulness of full-dispersion versions of traditional shallow-water models
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