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

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.