Near-Ordinary Periodic Waves of a Generalized Reaction-Convection-Diffusion Equation

We investigate near-ordinary periodic traveling wave solutions bifurcated from a family of ordinary periodic traveling wave solutions for a generalized reaction-convection-diffusion equation, especially its dependence on the nonlinear reaction. Using the Abelian integral method and the Chebyshev criteria, we find conditions for the existence and number of near-ordinary periodic wave solutions not only for the monotone case of the ratio of the Abelian integral as previous publications, but also for the non-monotone case. In a parameter region we provide a conjecture about the uniqueness of near-ordinary periodic traveling wave solutions for any degree of the nonlinear reaction and prove it up to degree four. The final simulations illustrate theoretical results numerically.