It is shown that the inverse scattering transform method for solving the Lax pair of given nonlinear evolution equation can be reduced to a kind of Riemann-Hilbert (RH) problem of meromorphic functions with respect to the complex spectral parameter. The RH problem is generally regular no matter solitons are involved or not. The linear singular integral equation associated with the RH problem has been derived, which is essential1y equivalent to the Gel fand-Levitan-Marchenko equation.Furthermore, the regtllar RH problem satisfied by the Sacklund transformation from a fundamental solution set of the eigenvalue equations of Lax pair to a new set has Fen given as well. The RH problem reduced from the inverse scattering transform is in fact a special case of that satisfied by the Backlund transformation.
Abstract
It is shown that the inverse scattering transform method for solving the Lax pair of given nonlinear evolution equation can be reduced to a kind of Riemann-Hilbert (RH) problem of meromorphic functions with respect to the complex spectral parameter. The RH problem is generally regular no matter solitons are involved or not. The linear singular integral equation associated with the RH problem has been derived, which is essential1y equivalent to the Gel fand-Levitan-Marchenko equation.Furthermore, the regtllar RH problem satisfied by the Sacklund transformation from a fundamental solution set of the eigenvalue equations of Lax pair to a new set has Fen given as well. The RH problem reduced from the inverse scattering transform is in fact a special case of that satisfied by the Backlund transformation.
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参考文献
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