Relaxation Oscillations And Homoclimic Orbits In Several Nonlinear Wave Equations And Biological Models

Based on geometric singular perturbation theory(GSPT),entry-exit function,Melnikov method and phase plane analysis method,etc.,this thesis studies nonlinear waves and relaxation oscillations as well as their bifurcations in several singular perturbation systems.More precisely,we study the existence of solitary wave solutions in a perturbed generalized Korteweg de Vries(KdV)equation,the existence of traveling waves and their classification as well as the persistence of solitary waves in a perturbed generalized Camassa-Holm(gCH)equation involving dual-power law nonlinearities,and the number of relaxation oscillations and the existence of homoclinic orbits in a singular perturbation predator-prey model with piecewise smooth functional response function.The thesis is divided into five chapters.Chapter 1 is the introduction of this thesis,in which,the research background,the related progresses and the structure of this thesis as well as the geometric singular perturbation theory are introduced.In Chapter 2,based on GSPT,phase plane analysis method and the “explicit”Melnikov method,we study the existence of solitary wave solutions in a perturbed generalized Korteweg de Vries(KdV)equation.Firstly,by the traveling wave transformation we obtain the associated traveling wave system,which is an ordinary differential equations(ODEs).Then by slow-fast decomposition and restricting the(slow)dynamics on the two-dimensional critical manifold,this ODE can be reduced to a planar Hamiltonian-perturbation system.We then analyze the global phase portraits of the unperturbed system associated with this Hamiltonian-perturbation system by using phase plane analysis method.Accordingly,we obtain the existence(persistence)of solitary wave solutions as well as its speed c by calculating the Melnikov integral explicitly on the basis of the explicit representation of homoclinic orbit.In Chapter 3,we study the existence of traveling waves as well as their classification and the persistence of solitary waves in a perturbed generalized Camassa-Holm(gCH)equation involving dual-power law nonlinearities.Firstly,by the traveling wave transformation the partial differential equation(PDE)is transformed into a singular traveling wave system with a singular straight line.After regularization we get the associated regular one,which is an integrable system.Then by using dynamical system approach we get the global phase portraits of the regular system and accordingly the singular one.By doing so,the existence of traveling waves as well as their classifications in this gCH equation involving dual-power law nonlinearities can be obtained.Then,under some special parameter conditions,we analyze the persistence of solitary waves,i.e.,the existence of homoclinic orbits in this gCH equation involving dual-power law nonlinearities under singular perturbation.In Chapter 4,based on GSPT and entry-exit function,we study the number of relaxation oscillations and the existence of homoclinic orbits in a singular perturbation predator-prey model with piecewise smooth functional response function.Firstly,by slow-fast separation we get the shape of the critical curves as well as the associated turning points.Then by taking the topology formed by the relative height of turning points and the positions of equilibriums into account,the number of relaxation oscillations and the existence of homoclinic orbits in this system are explored by using GSPT,entry-exit function and canard theory.In Chapter 5,we summary this thesis and give some problems to future work.