Monotonicity of Limit Wave Speed in the pgKdV Equation with Nonlinear Terms of Arbitrarily High Degrees

In this paper, the existence and number of periodic wave and solitary wave solutions of a pgKdV equation are established by applying the geometric singular perturbation theory and studying the limit cycle bifurcations of a perturbed hyperelliptic Hamiltonian system of arbitrarily high-order terms. For a general integer [Formula: see text], we provide a simple and rigorous proof for the monotonicity of the limit wave speed of traveling wave solutions. For the equivalent form of this equation, our results provide the complete answer to the open question posted by [ Yan et al. , 2014 ] and the conjecture proposed by [ Ouyang et al. , 2022 ].