We review new classes of exact solutions with functional parameters with constant asymptotic values at infinity of the Nizhnik-Veselov-Novikov equation and new classes of exact solutions with functional parameters of two-dimensional generalizations of the Kaup-Kupershmidt and Sawada-Kotera equations, constructed using the Zakharov-Manakov∂-dressing method. We present subclasses of multisoliton and periodic solutions of these equations and give examples of linear superpositions of exact solutions of the Nizhnik-Veselov-Novikov equation.
We use the Zakharov-Manakov∂-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) 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.
The classes of exactly solvable multiline soliton potentials and corresponding wave functions of two-dimensional stationary Schrödinger equation via -ץdressing method are constructed and their physical interpretation is discussed.
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