On The Scattering Approach And Inverse Scattering Problem For Dullin-Gottwald-Holm Equation

In this paper, we study the scattering approach theory and the inverse scattering problem for a new nonlinear dispersive shallow water wave equations ,named Dullin-Gottwald-Holm equation(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 directly 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.To begin with, DGH equation is converted into the classical Sturm-Liouville equation through the Liouville transformation so that the inverse scattering transform method is used to establish the inverse scattering equation of DGH equation as well as a series of solving equations. Thinking of the need of solution, we deal with the potential function by two methods, which may adapt to the different equations. The solving procedure of soliton solutions is presented on the base of the former theories. In the case of reflectionless potentials, the solving procedure of DGH equation amounts to solving a linear integral equation and a linear second-order ordinary differential equation. Thereby the known one-soliton solution and two-soliton solution of DGH equation are obtained in the simple parametric form according to the scattering data of DGH equation. However , the latter solution requires rather extensive use of mathematical packages, Mathematica and Matlab, in order to complete the construction of solutions. At the same time a few examples of the one-soliton solution profiles and two-soliton solution profiles at different times are presented, which show the familiar phase shift generated by the nonlinear interaction. In the following, the scattering approach of DGH equation is studied on the base of Lax pair of DGH equation and the Liouville transform. An initial potential, which is adjust to the inverse scattering problem, is obtained and an approach is also presented to prove the integrability of DGH equation for initial data of small amplitude. At last, the inverse scattering approach of DGH equation is given out. An alternative approach is proposed that provides a convenient algorithm for the inverse scattering problem. Moreover, another algorithm for the inverse scattering approach to DGH equation is obtained which is much simpler than the algorithm mentioned above. The applicability of these new approaches are also shown by means of examples.