INVERSE SCATTERING TRANSFORM AND REGULAR RIEMANN-HILBERT PROBLEM

Abstract

It is shown that the inverse scattering transform method to solve the Lax pair of given nonlinear evolution equation can be reduced to a kind of Riemann-Hilbert (R-H) problem of meromorphic functions with respect to the complex spectral parameter. The R-H problem is generally regular no matter whether the solitons are involved in it. The linear singular integral equation connected with the R-H problem has been derived, which is essencially equivalent to the Gel'fand-Levitan-Marchenko equation. Furthermore, the regular R-H problem satisfied by the Darboux-B?cklund transformation from a fundamental solution set of the eigenvalue equation of Lax pair to a new set has been given as well. The R-H problem reduced from the inverse scattering transform is in fact a special case of that satisfied by the Darboux-Backlund transformation.