反散射变换和正规Riemann-Hilbert问题

本文指出,非线性演化方程Lax表示的反散射交换可以化为关于亚纯函数的正规Riemann-Hilbert问题,并导出了相应的积分方程,后者在实质上与Gel’fand-Levitan-Marchenko方程等价。本文进而导出Lax表示的不同基本解组之间的Darboux-B?cklund变换所满足的正规Riemann-Hilbert问题。与反散射变换直接联系的正规Riemann-Hilbert问题是其特殊形式。 关键词：

INVERSE SCATTERING TRANSFORM AND REGULAR RIEMANN-HILBERT PROBLEM

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.
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