An integro-differential equation simulating a medium with strong spatial dispersion and hydrodynamic-type nonlinearities (the Whitham equation) is considered. A method is proposed for constructing the kernel of the integral term that makes it possible to qualitatively take into account the specifics of the dispersion laws of linear waves in media with spatial dispersion. The case where the kernel contains two independent parameters characterizing its amplitude and width is considered in detail. The dispersion laws of linear waves, as well as solutions in the form of solitary waves with limiting and small amplitudes, are obtained and analyzed. In particular, it is shown that the appropriate choice of parameters makes it possible to obtain the value of the tapering angle at the peak of a solitary wave with limiting amplitude on the surface of a fluid layer; this value is equal to the Stokes angle.
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Ursulov, A.V. Dispersion Laws, Nonlinear Solitary Waves, and Modeling of Kernels of Integro-Differential Equations Describing Perturbations in Hydrodynamic-Type Media With Strong Spatial Dispersion. Acoust. Phys. 66, 375–383 (2020). https://doi.org/10.1134/S1063771020040107
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DOI: https://doi.org/10.1134/S1063771020040107