Whitham modulation theory for the discontinuous initial-value problem of the generalized Kaup-Boussinesq equation

The Whitham modulation theory is developed to investigate the complete classification of solutions to discontinuous initial-value problem of the generalized Kaup-Boussinesq (KB) equation, which can model phenomenon of wave motion in shallow water. According to the dispersion relation, the generalized KB equation includes the generalized good-KB equation and generalized bad-KB equation, respectively. Firstly, the periodic wave solutions and the corresponding Whitham equations associated with the generalized bad-KB equation are given by Flaschka-Forest-McLaughlin approach. Secondly, the basic rarefaction wave structure and dispersive shock wave structure are described by analyzing the zero-genus and one-genus Whitham equations. Then the complete classification of solutions to Riemann problem of the generalized bad-KB equation is provided, and eighteen different cases are classified, including five critical cases. The distributions of Riemann invariants and the evolutions of self-similar states for each component are demonstrated in detail. It is shown that the exact soliton solution is in good agreement with the soliton edge of the modulated dispersive shock wave. Moreover, it is observed that the phase portraits in each case establish a consistent relationship with the behavior of the modulated solutions. Finally, for the generalized good-KB equation, anew type of discontinuous initial-value problem with constant-periodic wave boundaries is explored, and some novel modulated solutions with trigonometric shock waves are found. It is remarked that such trigonometric shock waves are absent in the generalized bad-KB equation because the small amplitude limits of the periodic waves are not trigonometric functions but constants. The results in this work reveal exotic wave-breaking phenomena in shallow water and provide a feasible way to investigate the discontinuous initial-value problem of nonlinear dispersive equations.