The Existence Of Traveling Wave Solutions For Two Nonlinear Differential Equations With Delays

The Belousov-Zhabotinskii system and the KdV equation are important nonlinear differential equations.The Belousov-Zhabotinskii system has a wide range of applications in biology and chemistry.The KdV equation is also often used in physics such as ion magnetic wave,non-resonant crystal vibration,and ion acoustic wave.This paper mainly discusses the Belousov-Zhabotinskii system and KdV equation with delays.By traveling wave transformations,the Belousov-Zhabotinskii system and KdV equation with delays are transformed into singular perturbation systems;thus we prove the existence of traveling wave solutions for Belousov-Zhabotinskii system and KdVequation by applying geometric singular perturbation theory(GSPT for short),implicit function theory,Melnikov integral and other methods.This paper is divided into fourchapters:Chapter 1 briefly introduces the research background and the main results of this paper.Chapter 2 mainly introduces the GSPT and its basic knowledge.Chapter 3 discusses the Belousov-Zhabotinskii system with delay.By the traveling wave transformation,we can transform the partial differential system into a singular perturbation system.And then,by applying the GSPT,we can obtain a local invariant manifold.At this moment,we discuss the system that is limited on the local invariant manifold,and obtain the existence of the heteroclinic orbit by Fredholm alternative theorem.According to the relationship between the traveling wave solution and the heteroclinic orbit,we obtain that there is a traveling wave solution in the original Belousov-Zhabotinskii system.Chapter 4 considers the periodic orbits and solitary wave in a KdV equation with delay.We prove the existence of solitary wave solutions and periodic solutions by applying GSPT,implicit function theorem and Melnikov integral method.Moreover,by applying the Abelian integral theory,we can describe the monotonicity and range of the periodic orbit's wave speed c,and the monotonicity and range of the periodic orbit's wavelength l with respect to the wave speed c.