Geometric Singular Perturbation Analysis Of Several Slow-fast Systems

In this paper,we study two nonlinear partial differential equations: Degasperis-Procesi equation with distributed delay and neuron model with conductance-resistance symmetry.These are both important mathematical models for studying physical and biological phenomena.In this paper,we mainly prove the existence of solitary wave solutions of these two equations by using the geometric singular perturbation theory and invariant manifolds theory,phase diagram analysis and blow-up technique.In Chapter 2,we give some analysis tools that need to be used in this paper: Fenichel's singular perturbation theory,Melnikov's method,exchange lemma and blow-up technique.In Chapter 3,we consider the Degasperis-Procesi equation,which is an approximation to the incompressible Euler equation in shallow water regime.First we provide the existence of solitary wave solutions for the original Degasperis-Procesi equation.Then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the geometric singular perturbation theory and invariant manifold theory.According to the relationship between solitary wave and homoclinic orbit,the Degasperis-Procesi equation is transformed into the slow-fast system by using the traveling wave transformation.It is proved that the perturbed equation also has a homoclinic orbit,which corresponds to a solitary wave solution of the delayed Degasperis-Procesi equation.In Chapter 4,we consider a recently proposed the neuron model with conductanceresistance symmetry which is similar to the Hodgkin-Huxley equation,referred to as CRSN equation.The model was proposed by Professor Deng in 2019.The system to start is to assume that two modelers must obtain the same model when one models the conductances of ion channels and the other models the channels' resistances.To the best of our knowledge,the traveling wave solutions of the neural model with conduction-resistance symmetric and its reduction model have not been studied,and ours is the first.In this chapter,we will consider a 2-dimensional reduction model of the CRSN equation in a propagated action potential.Firstly,we give the derivation of the neuron model with conductance-resistance symmetry in action potential.Secondly,since the system is not a classical slow-fast system,and it will change in a special regional time scale,we prove the existence of solitary wave solution for neuron model with conductance-resistance symmetry by using the method of phase diagram analysis.Finally,we obtain the limit orbit of the solitary wave solution by using the geometric singular perturbation theory,exchange lemma and the blow-up technique in different regions..In the last chapter,we summarize the main results of this dissertation and propose some questions for further study.