The Number Of Zeros Of Abel Integrals And Their Applications

In this paper,we study the number of zeros of some Abelian integrals and their application to the number of periodic traveling wave solution of nonlinear wave equation.In the first chapter,we introduce the related background of the number of zeros of Abelian integral of hyperelliptic systems and traveling wave solutions of nonlinear wave equations,as well as our results.In the second chapter,we study the exact bound on the maximal number of limit cycles by Poincar’e bifurcation for a class of seventh order symmetric hyperelliptic Hamiltonian systems under small perturbation.According to Poincar’e bifurcation theory,the number of maximum limit cycles of the system can be determined by the maximal number of zeros of Abelian integrals.We mainly consider the seventh order symmetric hyperelliptic Hamiltonian system with two hyperbolic saddles and two nilpotent cusps.Using algebraic methods such as Cheyshev system theory,calculation method with the help of relevant symbols,determining whether there are zeros in related semi-algebraic systems.By analyzing the ratio of Abelian integral near the center and heteroclinic loop,the existence of the maximal number of zeros is obtained.Finally,we obtain the maximal number of zeros of Abelian integral corresponding to Hamiltonian system.In the third chapter,we study the existence and coexistence of kink waves and periodic waves for a class of quintic perturbed BBM equations.The main method is to transform the nonlinear wave equation into a singularly perturbed dynamic system by traveling wave transformation,constructing perturbed Hamiltonian system on critical manifolds,the corresponding Abelian integral is obtained.Using Chebyshev system and asymptotic expansion method,the number and distribution of zeros of Abelian integral are obtained.Then,we obtain the number and distribution of periodic waves and kink waves.Using the geometric and algebraic methods,we get the maximal number of zeros of Abelian integrals by studying the maximal number zeros of zeros of the second derivative of Abelian integral.Finally,we get that the maximal upper bound of number of periodic traveling wave solutions is 2.In the last chapter,we summarize our research results,and put forward some suggestions for the future research work.