Nonlinear Waves In Sine-Gordon Equations With Slowly Varying Parameters And Heterogeneities

By combining Fenichel's geometric singular perturbation theory,Melnikov func-tion method and phase plane analysis,this paper studies the existence of single-pulse/multiple-pulse travelling solutions in sine-Gordon equations with slowly vary-ing parameters and heterogeneities.The thesis is consisted of five chapters:Chapter 1 is the introduction,in which,we introduce Fenichel's geometric sin-gularly perturbed theory,sine-Gordon equation and its perturbations as well as several typical heterogeneities,and finally,we give the structure of this thesis.In chapter 2,we study the existence of single-pulse travelling front solutions to a sine-Gordon equation with slo,wly varying parameters.By slow-fast decomposition,we get the layer and reduced systems as well as their global dynamics.Then we introduce Melnikov function to measure the distance between the stable and unstable manifolds of the slow manifold,and control the Take-off and Touch-down curves to respectively intersect with the stable and unstable manifolds of the saddles on two slow manifolds transversally.Hence we get the singular heteroclinic orbits with transversality.Accordingly,we get the existence of heteroclinic orbits of the full system by perturbing the singular heteroclinic orbits.In chapter 3,we study the existence of pinned travelling front solut,ions to a sine-Gordon equation with jump-like and bump-like heterogeneities.In fact,the heterogeneous sine-Gordon equation is a piecewise smooth dynamic system,which is non-autonomous.By phase plane analysis,we get,the exist.ence of pinned solutions under the perturbations of heterogeneities and determine the positions of pinning.In chapter 4,we study the existence of pinned single-pulse/multiple-pulse trav-elling front solutions to a sine-Gordon equation with strongly localized heterogeneity of Dirac delta type.By phase plane analysis and matching,we study the existence and the types of nonlinear waves under the perturbation of single-heterogeneity or two heterogeneities respectively.In chapter 5,the summaries on this thesis are given.