A nonlocal shallow-water model arising from the full water waves with the Coriolis effect

Abstract

In the present study a mathematical model of long-crested water waves propagating mainly in one direction with the effect of Earth's rotation is derived by following the formal asymptotic procedures. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations and has a formal bi-Hamiltonian structure. Its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Our refined analysis is approached by applying the method of characteristics and conserved quantities to the Riccati-type differential inequality.

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Cited by in corpus (6)

• Well-posedness, travelling waves and geometrical aspects of generalizations of the Camassa-Holm equation
• Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations
• Wave breaking for shallow water models with time decaying solutions
• Integrability, existence of global solutions and wave breaking criteria for a generalization of the Camassa-Holm equation
• Well-Posedness of a kind of the free surface equation of shallow water wave
• Existence and uniqueness of global weak solutions to a generalized Camassa-Holm equation