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Darboux transformation for the nonstationary Schrödinger equation. (Russian. English summary) Zbl 1094.37037

Zap. Nauchn. Semin. POMI 317, 94-104 (2004); translation in J. Math. Sci., New York 136, No. 1, 3580-3585 (2006).
The author of this very interesting paper studies the nonlinear nonstationary Schrödinger equation (NtSH for short), \[ i\Psi_t=-\Psi_{xx}+\kappa (\Psi , \overline\Psi )\Psi , \] where \(\kappa (\Psi , \overline\Psi )\equiv W(x,t)\) is a certain real function of both independents \(x\) (space) and \(t\) (time). It is known that the above stated mathematical model belongs to the class of completely integrable systems. The Lax method is applied in the Heisenberg picture. Consideration of Lax pairs of equations for a single function leads to a compatibility condition that is, in general, a nonlinear equation for their coefficients, called potentials. This means that the nonlinear equation is invariant with respect to the potential part of the Darboux transformation.
The author uses the Darboux transformation that could be applied many times in order to obtained exact solutions to the problem under consideration. Thus the Darboux transformation gives the symmetry for the nonlinear equation with constraints. Some examples of the construction of exact solutions are discussed.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81R12 Groups and algebras in quantum theory and relations with integrable systems
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
81U40 Inverse scattering problems in quantum theory
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
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