| 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 Ehrnstr{\o}m, Groves \& Wahl\`{e}n 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. |
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