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<p style='text-indent:20px;'>The integrability, classification of traveling wave solutions and stability of exact solutions for the generalized Kaup-Boussinesq equation are studied by prolongation structure technique and linear stability analysis. Firstly, it is proved that the generalized Kaup-Boussinesq equation is completely integrable in sense of having Lax pair. Secondly, the complete classification of exact traveling wave solutions of the generalized Kaup-Boussinesq equation are given and a family of exact solutions are proposed. Finally, the stability of these exact solutions are investigated by linear stability analysis and dynamical evolutions, and some stable traveling wave solutions are found. It is shown that the results of linear stability analysis are in excellent agreement with the results from dynamical evolutions.</p>

The complete classification of solutions to the Riemann problem of the good Jaulent–Miodek equation is investigated by Whitham modulation theory. The one-phase and two-phase periodic wave solutions and the corresponding Whitham equations are derived based on the Lax pair of the good Jaulent–Miodek equation and the finite-gap integration method. In particular, the N-phase periodic wave solutions are proposed by algebro-geometric approach. Then the basic wave structures of rarefaction waves and DSWs are proposed analytically and graphically, which makes it possible to establish the classification of all the possible wave patterns evolving from initial discontinuities. The asymptotic results given by Whitham modulation theory are in excellent agreement with direct numerical simulations. Finally, a detailed description of the shallow-water dam break problem is demonstrated to find the possible physical significance of the wave patterns found in this work.