Study Of The Existence Of Traveling Wave Solutions For A Perturbed Combined Double-Dispersive Equation And A Keller-Segel Equation

In this thesis,we study the global existence and uniqueness of periodic wave for a perturbed combined double-dispersive equation,and investigate the existence of traveling pulse solutions for a Keller-Segel equation with small cell diffusion and nonlinear stimulation.In Chapter 1,we introduce some backgrounds on two kinds of PDEs,a short description on our two modes.In Chapter 2,we study the periodic wave propagation phenomenon in elastic waveguides modeled by a combined double-dispersive partial differential equation(PDE).The traveling wave ansazt transforms the PDE model into a perturbed integrable ordinary differential equation(ODE).The global bifurcation theory is applied for the perturbed ODE model to explore the existence and uniqueness of the limit cycle,which corresponds the periodic traveling wave for the PDE model.The main tool is the Abelian integral taken from Poincaré bifurcation theory.Simulation is carried out to verify the theoretical result.In Chapter 3,we study traveling waves for a Keller-Segel equation with weak cell diffusion under nonlinear cellular stimulation.We transform the Keller-Segel equation into a dynamical system with a singular perturbation,then investigate the dynamics on a normally hyperbolic invariant manifold by applying geometric singular perturbation theory.A compact invariant region is constructed to bound the unstable manifold of a homogeneous state aiming at exploring the connections between two states.The existence on the pulse for the bacteria density and kink front for chemical density is explored.The exact conditions for oscillatory tail and monotonic decay in the front are obtained.Two examples are also presented to illustrate our results,and the range of the wave speed is determined numerically for each system.Finally,we discuss the(in)stability of the pulse by analyzing the spectrum of the related differential operator for the original partial differential equation.In the last Chapter,the main work of the thesis is summarized,and some prospects for future research work are put forward.