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Matrix solutions of the equation\(\mathfrak{B}U_t = - U_{xx} + 2U^3 + \mathfrak{B}[U_x ,U] + 4cU:\) extension of the method of the inverse scattering problem

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Abstract

Complex solution matrices of the nonlinear Schrödinger equation \(\mathfrak{B}U_t = - U_{xx} + 2U^3 + \mathfrak{B}[U_x ,U] + 4cU:\) are found and the method of the inverse scattering problem is subjected to a natural extension. That is, for the nonself-conjugateL-A Lax doublet that arises for this equation, the presence of chains of adjoint vectors for the operatorL is taken into account by means the corresponding normed chains. A uniqueness theorem for the Cauchy problem for the above Schrödinger equation is obtained. Here\(\mathfrak{B} = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right),[M,N] = MN - NM\), and c is a parameter.

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  1. Institute of Applied Problems of Mechanics and Mathematics, Academy of Sciences of Ukraine, Lvov

    I. -P. P. Syroyid

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  1. I. -P. P. Syroyid
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Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1264–1275, September, 1992.

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Syroyid, I.P.P. Matrix solutions of the equation\(\mathfrak{B}U_t = - U_{xx} + 2U^3 + \mathfrak{B}[U_x ,U] + 4cU:\) extension of the method of the inverse scattering problem. Ukr Math J 44, 1156–1166 (1992). https://doi.org/10.1007/BF01058378

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  • DOI: https://doi.org/10.1007/BF01058378

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