Periodic waves in the pgKdV equation with two arbitrarily high-order nonlinearities

In this paper, the existence and number of periodic wave solutions in a perturbed generalized KdV equation of high-order with weak backward diffusion and dissipation effects are studied. These can be converted into studying the periodic waves on a manifold via geometric singular perturbation theory . By using bifurcation theory and analyzing the number of real zeros of some linear combination of Abelian integrals whose integrand and integral curve both have two arbitrarily high-order terms, we prove the persistence of periodic waves with certain wave speeds under small perturbation. The persistence of periodic waves for any energy parameter in an open interval and sufficiently small parameter is also established. Furthermore, the monotonicity of the limit wave speed is given, and the upper and lower bounds of limit wave speed are obtained. It is the first time to prove the existence of periodic waves in this kind of equation with two arbitrarily high-order nonlinearities.