New exact solutions with functional parameters of the Nizhnik—Veselov—Novikov equation with constant asymptotic values at infinity

Abstract

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.

New exact solutions of two-dimensional integrable equations using the $\bar \partial $ -dressing method

“…It can be shown that the potentiality condition can be satisfied by the choice of the functional parameters in the spectral representation [26] …”

“…, N, are some constants. Reality condition (2.5) leads to additional restrictions on p k and q k [26] …”

“…Other examples of exact solutions with functional parameters for both versions of the NVN equation were obtained in [26].…”

“…The solutions of NVN equation (1.1) with constant asymptotic values at infinity u(ξ, η, t) =ũ(ξ, η, t) + u ∞ =ũ(ξ, η, t) − , whereũ(ξ, η, t) → 0 as ξ 2 + η 2 → ∞, and with the functional parameters α k (ξ, η, t) and β k (ξ, η, t) defined in (1.11) and (1.8) are given by the general determinant formula [26] …”

“…26) with some real constants v k . General determinant formula (1.14) in case (3.23) under consideration gives another class of exact solutions u with functional parameters of 2DSK equation (1.3).…”