Related Study On The Eigenvalues Of The Generalized Periodic Camassa-Holm Equations

By understanding the spectrum,we can not only understand the dynamic character-istics of linear system itself,but also understand related dynamical behavior of nonlinear system.In recent years,the famous Camassa-Holm shallow water wave equation with periodic boundary conditions of the spectrum problem has attracted extensive attention of many scholars and experts.However,for the spectral problem of generalized Camassa-Holm equations with periodic boundary conditions,no relevant work has been done.In this paper we are concerned with the spectral problem for the periodic generalized Camassa-Holm equations（?）This paper is divided into three chapters.In chapter 1,we first introduce the spectral problem and the inverse spectral problem corresponding to the periodic Camassa-Holm equation.Then the latest progress of the generalized periodic Camassa-Holm equation is introduced,and the main work of this paper is briefly described.In chapter 2,we first introduce the preliminary knowledge of the convergence of func-tions in the weak topological sense of Lebesgue space L1[0,T].Then we use a dynamical approach to prove that（0.0.1）with the separated boundary condition y1（0）sinα+y2（0）cos α=0,y1（T）sinβ+y2（T）cosβ=0,has infinity real eigenvalues {λ_k（m）} satisfying …<λ_-2（m）<λ_-1（m）<λ₀（m）<λ₁（m）<λ₂（m）<… and limk→±∞λ_k（m）=±∞.Moreover,each eigenvalue λ_k（m）is simple.Then we are able to construct the periodic eigenvalues and anti-periodic eigenvalues by {λ_k（m）}.The eigenvalues of Equation（0.0.1）satisfying the periodic boundary condition and the eigenvalues satisfying the anti-periodic boundary condition can be written as a sequence of real numbers as follows （?）where （?）_k（m）and （?）_k（m）are eigenvalues of（0.0.1）satisfying the periodic boundary for k being even,and （?）_k（m）and （?）_k（m）are eigenvalues of（0.0.1）satisfying the anti-periodic boundary for k being odd.Finally,we prove that as nonlinear functionals of potentials,eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space L¹/[0,T].In chapter 3,we find the estimates of the eigenvalues when the L1 norm of potentials is given,based on an application of trace formulas.