The Inverse Spectral Transform Of Some Continuous And Discrete Integrable Equations

The inverse spectral transform method is used to investigate the problem of several continuous and discrete integrable equations.Their soliton solutions are obtained.The problems considered in this thesis involve TD equation with non-zero boundary condition,two-component generalized Ragnisco-Tu equation with zero boundary condition and Tzitzeica equation with zero boundary condition.The key procedure in the inverse spectral transform is the spectral analysis for the linear spectral problem of the nonlinear integrable equation.The difficult point in this work is that the associated spectral spaces of some equations involve multi-sheet Riemann surfaces.So we need to change the spectral variables and to make sure the spectral space to be certain Riemann spheres.Then,in the new spectral spaces,we introduce the elementary matrix solutions ?±(also be called Jost functions)of the associated linear spectral problems,and certain scattering matrix S(k).By use of the properties of Green function,we give the analysis of the associated integral equation,and discuss analytical properties of the Jost functions and elements of the scattering matrix.Furthermore,we consider the asymptotic behaviors of the Jost function at certain singularities.Another impor-tant procedure in the inverse spectral transform is to reconstruct the potentials,and to find the explicit solutions.To do this,we need to define new sectional-ly meromorphic(or analytic)function in order to construct the Riemann-Hilbert problem.By virtue of the normalization procedure or Cauchy projector,the po-tential is reconstructed from the scattering data.At last,the explicit solutions,including solitons,of the integrable equations are given under the condition of reflectionless potential.The four problems of the integrable equations share one common feature,the zeroes of the scattering coefficients are single.It is noted that the zeros of scattering coefficients in chapter 2,4 and 5 only exist in their analytic region,and there is no zero at the boundary of the domain.While the zeros in chapter 3 exist in their analytic domain and on their boundary.Therefore,in the process of finding soliton solutions,the construction of the RH problem will be dealt with in different ways.