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Reductions and exact solutions of nonlinear multidimensional Schrödinger equations. (English. Russian original) Zbl 0746.35041

Theor. Math. Phys. 87, No. 2, 488-498 (1991); translation from Teor. Mat. Fiz. 87, No. 2, 220-234 (1991).
Summary: Using the canonical decomposition of an arbitrary subalgebra of the orthogonal algebra \(AO(n)\), we describe the maximal subalgebras of rank \(n\) and \(n-1\) of the extended isochronous Galileo algebra, and also the maximal subalgebras of rank \(n\) of the generalized extended classical Galileo algebra \(A\tilde G(1,n)\), the extended special Galileo algebra \(A\tilde G(2,n)\), and the extended complete Galileo algebra \(A\tilde G(3,n)\). Using the subalgebras of rank \(n\), we construct ansatzes that reduce multidimensional Schrödinger equations to ordinary differential equations. Exact solutions of the Schrödinger equations are found from the solutions of the reduced equations.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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References:

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