About simple nonlinear and linear superpositions of special exact solutions of VeselovNovikov equation
Abstract
New exact solutions, nonstationary and stationary, of VeselovNovikov (VN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u{sup (n)}, n= 1, HorizontalEllipsis , N are constructed via Zakharov and Manakov {partial_derivative}dressing method. Simple nonlinear superpositions are represented up to a constant by the sums of solutions u{sup (n)} and calculated by {partial_derivative}dressing on nonzero energy level of the first auxiliary linear problem, i.e., 2D stationary Schroedinger equation. It is remarkable that in the zero energy limit simple nonlinear superpositions convert to linear ones in the form of the sums of special solutions u{sup (n)}. It is shown that the sums u=u{sup (k{sub 1})}+...+u{sup (k{sub m})}, 1 LessThanOrSlantedEqualTo k{sub 1} < k{sub 2} < HorizontalEllipsis < k{sub m} LessThanOrSlantedEqualTo N of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schroedinger equation and can serve as model potentials for electrons in planar structures of modern electronics.
 Authors:
 Novosibirsk State Technical University, Karl Marx prosp. 20, Novosibirsk 630092 (Russian Federation)
 Publication Date:
 OSTI Identifier:
 22162823
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 54; Journal Issue: 3; Other Information: (c) 2013 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ELECTRONS; ENERGY LEVELS; EXACT SOLUTIONS; NONLINEAR PROBLEMS; PERIODICITY; SCHROEDINGER EQUATION; SOLITONS; WAVE PROPAGATION
Citation Formats
Dubrovsky, V. G., and Topovsky, A. V.. About simple nonlinear and linear superpositions of special exact solutions of VeselovNovikov equation. United States: N. p., 2013.
Web. doi:10.1063/1.4795132.
Dubrovsky, V. G., & Topovsky, A. V.. About simple nonlinear and linear superpositions of special exact solutions of VeselovNovikov equation. United States. doi:10.1063/1.4795132.
Dubrovsky, V. G., and Topovsky, A. V.. 2013.
"About simple nonlinear and linear superpositions of special exact solutions of VeselovNovikov equation". United States.
doi:10.1063/1.4795132.
@article{osti_22162823,
title = {About simple nonlinear and linear superpositions of special exact solutions of VeselovNovikov equation},
author = {Dubrovsky, V. G. and Topovsky, A. V.},
abstractNote = {New exact solutions, nonstationary and stationary, of VeselovNovikov (VN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u{sup (n)}, n= 1, HorizontalEllipsis , N are constructed via Zakharov and Manakov {partial_derivative}dressing method. Simple nonlinear superpositions are represented up to a constant by the sums of solutions u{sup (n)} and calculated by {partial_derivative}dressing on nonzero energy level of the first auxiliary linear problem, i.e., 2D stationary Schroedinger equation. It is remarkable that in the zero energy limit simple nonlinear superpositions convert to linear ones in the form of the sums of special solutions u{sup (n)}. It is shown that the sums u=u{sup (k{sub 1})}+...+u{sup (k{sub m})}, 1 LessThanOrSlantedEqualTo k{sub 1} < k{sub 2} < HorizontalEllipsis < k{sub m} LessThanOrSlantedEqualTo N of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schroedinger equation and can serve as model potentials for electrons in planar structures of modern electronics.},
doi = {10.1063/1.4795132},
journal = {Journal of Mathematical Physics},
number = 3,
volume = 54,
place = {United States},
year = 2013,
month = 3
}

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