Solitonic combinations and commuting nonselfadjoint operators

In this paper, solutions of different nonlinear differential equations are obtained using the connection between the theory of solitons and the theory of commuting nonselfadjoint operators, established by M.S. Livšic and Y. Avishai in [15] and based on the Marchenko method for solving nonlinear differential equations. n-tuples of commuting nonselfadjoint bounded linear operators, when one of them belongs to a larger class of nondissipative operators, are used. Preliminary results are obtained with the help of the properties of this larger class of operators—couplings of dissipative and antidissipative operators with real absolutely continuous spectra (whose triangular model and corresponding nondissipative curves have been introduced and investigated by the authors in [5]). The preliminary results concern the application of this connection for the considered n-tuples of commuting operators. A generalized open system, corresponding to collective motions, is introduced. Solutions of the nonlinear Schrödinger equation, the Heisenberg equation, the Sine-Gordon equation, the Davey–Stewartson equation are obtained, using the generalized open systems and the matrix wave equations for appropriate pairs and triplets of commuting nonselfadjoint operators.