Geodesic
New exact solutions with functional parameters of the~Nizhnik--Veselov--Novikov equation with constant asymptotic values at infinity
Teoretičeskaâ i matematičeskaâ fizika, Tome 165 (2010) no. 2, pp. 272-294

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We use the Zakharov–Manakov $\bar{\partial}$-dressing method to construct new classes of exact solutions with functional parameters of the hyperbolic and elliptic versions of the Nizhnik–Veselov–Novikov equation with constant asymptotic values at infinity. We show that the constructed solutions contain classes of multisoliton solutions, which at a fixed time are exact potentials of the perturbed telegraph equation {(}the perturbed string equation{\rm)} and the two-dimensional stationary Schrödinger equation. We interpret the stationary states of a microparticle in soliton-type potential fields physically in accordance with the constructed exact wave functions for the two-dimensional stationary Schrödinger equation.
Keywords: Nizhnik–Veselov–Novikov equation, solution with functional parameters, perturbed telegraph equation, two-dimensional stationary Schrödinger equation, transparent potential.
Mots-clés : soliton 
@article{TMF_2010_165_2_a5,
     author = {V. G. Dubrovskii and A. V. Topovsky and M. Yu. Basalaev},
     title = {New exact solutions with functional parameters of {the~Nizhnik--Veselov--Novikov} equation with constant asymptotic values at infinity},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {272--294},
     publisher = {mathdoc},
     volume = {165},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2010_165_2_a5/}
}
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V. G. Dubrovskii; A. V. Topovsky; M. Yu. Basalaev. New exact solutions with functional parameters of the~Nizhnik--Veselov--Novikov equation with constant asymptotic values at infinity. Teoretičeskaâ i matematičeskaâ fizika, Tome 165 (2010) no. 2, pp. 272-294. http://geodesic.mathdoc.fr/item/TMF_2010_165_2_a5/