Traveling Wave Solutions In Several Singularly Perturbed Reaction-advection-diffusion Systems

By combining geometric singular perturbation theory with the technique of generalized rotation vector field,phase plane analysis and Melnikov method,etc.,this thesis studies the existence and asymptotic behaviors of traveling wave solutions in several singularly perturbed reaction-convection-diffusion systems.The thesis is divided into five chapters,which are as follows:Chapter 1 is the introduction,in which,the research background of this thesis,geometric singular perturbation theory,generalized rotation vector field theory as well as the structure of the thesis are introduced.In Chapter 2,based on geometric singular perturbation theory and generalized rotation vector field,we study the local patterns in a Klausmeier model via a more geometric method.Firstly,a three-dimensional singularly perturbed ODE system with two fast variables and a slow variable is obtained.Then the dynamical behaviors of the associated slow and fast limit systems are analyzed in detail.More precisely,we introduce the generalized rotation vector field theory to show the birth of heteroclinic orbits of the layer system.By doing so,we can select the proper fronts and backs and match them with the slow orbits to construct all the possible singular homoclinic and heteroclinic orbits.Finally,the singular homoclinic/heteroclinic orbits are classified and are perturbed,and finally the existence of traveling wave patterns in the Klausmeier model are obtained.In Chapter 3,we are concerned with the travelling pulse patterns in a generalized Klausmeier-Gray-Scott(g KGS)model with higher-order nonlinearity by employing geometric singular perturbation theory and an“explicit” Melnikov method.We pay our attention to reveal how the nonlinearity affects the pulse solutions.By introducing a suitable set of rescaling on variables and parameters,the problem is changed into a slow-fast setting which can then be analyzed by GSPT.By computing the Melnikov integral and determining the so-called Take-off and Touch-down curves explicitly,the critical parameter conditions for the existence of travelling single-pulse/multi-pulse patterns in this g KGS model are obtained,and it is found that they are tightly related to the power of higher-order nonlinearity,which are expressed in a “double factorial” way.From the obtained critical parameter conditions,the quantitative influence from the higher-order nonlinearity is then knownIn Chapter 4,based on geometric singular perturbation theory,we study the existence of steady-state wave and traveling wave patterns of a vegetation-water-toxic compound coupled model with a higher-order nonlinearity.Firstly,by introducing the traveling wave coordinate transformation and the rescalling on parameters and variables,a five-dimensional singularly perturbed ODE system is obtained.Then the dynamical behaviors of the slow and fast limiting systems are analyzed and are matched to form the singular orbits.Finally,we prove the existence of steady-state wave and traveling wave patterns by perturbing these singular orbits.We find that the steady-state and traveling wave solutions appear under different parameter scale transformations.Chapter 5 is the summary on this thesis.Some unsolved problems are also given in this section.