Monotonicity Of Limit Wave Speed Of Traveling Wave Solutions For A Perturbed Generalized KdV Equation

This paper mainly uses the geometric singular perturbation theory to study the existence of the traveling wave solution of the perturbed generalized KdV equation under the conditions of n=2 and n=3.And the limiting velocity of the traveling wave solution of the equation is described as the ratio of the Abel integral,which proves that the limiting wave velocityc₀（h）decreases monotonically.The value range of the limit wave speed and the relationship between the wave speed and the traveling wave wavelength are also given.When n=4,it is proved that the limit wave speed is monotonically decreasing,and the numerical simulation is carried out with mathematical software.Our results partially answer an open question posed by Yan et al.in[Math.Model.Anal.,19（2014）,pp.537-555].This article consists of six chapters,arranged as follows:The first chapter introduces the research background,current situation and main research contents of the traveling wave solution of the perturbed generalized KdV equation.In the second chapter,the related knowledge of plane Hamiltonian system and geometric singular perturbation theory is given.The third,fourth and fifth chapters respectively introduce the monotonicity of the limiting wave velocity of the traveling wave solution of the perturbed generalized KdV equation in the case of n=2,3,4.First,the velocity of the traveling wave solution of the equation is expressed as the ratio of the Abel integral,and the Picard-Fuchs equation is used for analysis,and then it is transformed into a Riccati equation for the Abel integral.Finally,it is concluded that the limit velocity is monotonically decreasing,and given its value range.Then,the existence of the traveling wave solution of the equation is proved by analyzing the low-dimensional fast-slow system by using the geometric singular perturbation theory.The limiting conditions of the limit wave velocity are obtained.Finally,some properties of the periodic wave solution are given when the equation n=2,3,and the relationship between the wave speed and the wavelength is obtained.Chapter 6 summarizes the research content of this paper and looks forward to future research directions.