Bilinear Problem Of Dullin-Gottwald-Holm Equation And Spectral Decomposition Of A General Chebyshev Map

In this paper, there is two parts. The first part: we study the Hirota problem for a new nonlinar dispersive shallow water wave equations, named Dullin-Gottwald-Holm (i.e. DGH equation). DGH equation is the 1+1 quadratically nonlinear equation for unidirectional water waves,which was derived by Dullin,Gottwald and Holm,by using asymptotic expansions direcly in the Hamiltonian for Euler's equations in the irrotational incompressible flow of a shallow layer of inviscid fluid moving under the in fluence of gravity as well as surface tension. It is a completely-integrable equation. The second part: we study the spectral decomposition of a general Chebyshev maps.To begin with, we use reciprocal transformation to map DGH equation to associaed DGH equation. Use samiliar reciprocal transformation on Lax pairs of DGH equation, we obtain a new integrable system through introducing appropriate potential function. This integrable system includes variable u(x,t) .We get a integrable system of (y,t) space through eliminating the variable of the integrable system. The integrable system has some relation with BBM equation. The bilinear form of BBM equation has known, so we can similiarly get the bilinear form of associated DGH equation, then we can get the bilinear form of DGH equation and 1-soliton by a approciate method.In the following, we discuss the spectral decomposition of a general Chebyshev maps. In order to construct the spectral decomposition of the Frobenius-Perron operator for a general Chebyshev maps We define a suitable dual pairs or rigged Hilbert space, which provides mathematical meaning for the spectral decomposition. Construct the spectral decompositions of the family of Tent maps under the Frobenius-Perron operator. We can get the eigenvalues and eigenvectors of the Tent maps. Through the topological equivalence of transformations, we can take the general Chebyshev maps into the Tent maps, then we can get the eigenvalues and eigenvectors of the general Chebyshev maps.