This paper focus on the existence and uniqueness of periodic waves for a BBM equation with local strong generic delay convection and weak diffusion. By analyzing the corresponding Hamiltonian system, we aim to obtain the existence of periodic orbit by constructing a locally invariant manifold according to geometric singular perturbation theory. Chebyshev criteria is applied to investigate the ratio of Abelian integrals. We prove the existence and uniqueness of periodic wave solution with sufficiently small perturbation parameter. Moreover, the upper and lower bounds of the limiting wave speed are given. Keywordsdelayed BBM equation • geometric singular perturbation theory • periodic wave • Abelian integrals Mathematics Subject Classification (2020) 34C25 • 34C60 • 37C27 1 IntroductionTraveling wave solutions plays an important role in many mathematically modelled phenomena. It can be applied in comparison principles and characterize the long-term behaviour in numerous situations in conformance with

This paper focus on the existence and uniqueness of periodic waves for a BBM equation with local strong generic delay convection and weak diffusion. By analyzing the corresponding Hamiltonian system, we aim to obtain the existence of periodic orbit by constructing a locally invariant manifold according to geometric singular perturbation theory. Chebyshev criteria is applied to investigate the ratio of Abelian integrals. We prove the existence and uniqueness of periodic wave solution with sufficiently small perturbation parameter. Moreover, the upper and lower bounds of the limiting wave speed are given. Mathematics Subject Classification (2020) 34C25 · 34C60 · 37C27

In this paper, we consider the KP-MEW(3,2) equation by the bifurcation theory of dynamical systems when integral constant is considered. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. The phase portrait for c < 0 , 0 < c < 1 , and c > 1 is drawn. Exact parametric representations of periodic peakon solutions and smooth periodic solution are presented.

<abstract><p>A perturbed MEW equation including small backward diffusion, dissipation and nonlinear term is considered by the geometric singular perturbation theory. Based on the monotonicity of the ratio of Abelian integrals, we prove the existence of periodic wave on a manifold for perturbed MEW equation. By Chebyshev system criterion, the uniqueness of the periodic wave is obtained. Furthermore, the monotonicity of the wave speed is proved and the range of the wave speed is obtained. Additionally, the monotonicity of period is given by Picard-Fuchs equation.</p></abstract>

In the presented paper, a generalized nonlinear Schr o dinger equation without delay convolution kernel and with special delay convolution kernel is investigated. By using the geometric singular perturbation theory, the existence of traveling wave fronts is proved. Firstly, we show that such traveling wave fronts exist without delay by non-Hamiltonian qualitative analysis. Then, for the generalized nonlinear Schr o dinger equation with a special local strong delay convolution kernel, the desired heteroclinic orbit is obtained by using the Fredholm theory.