About linear superpositions of special exact solutions of Veselov-Novikov equation

TL;DR

This paper constructs new exact solutions for the Veselov-Novikov equation using superpositions of special solutions, including solitons and periodic solutions, which also serve as transparent potentials for 2D Schrödinger equations.

Contribution

It introduces a method to generate linear superpositions of exact solutions of the Veselov-Novikov equation, expanding the class of known solutions and their applications.

Findings

Superpositions include line solitons and periodic solutions.

Sum of arbitrary subsets of solutions are also solutions.

Solutions serve as transparent potentials for 2D Schrödinger equation.

Abstract

New exact solutions, nonstationary and stationary, of Veselov-Novikov (VN) equation in the forms of linear superpositions of arbitrary number of exact special solutions $u^{(n)}$, $n=1,...,N$ are constructed via $\bar\partial$-dressing method in such a way that the sums $u= u^{(k_1)}+...+ u^{(k_m)}$, $1\leqslant k_1<k_2<...<k_m\leqslant N$ of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented linear superpositions include as superpositions of special line solitons with zero asymptotic values at infinity and also superpositions of special plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schr\"{o}dinger equation and can serve as model potentials for electrons in planar structures of modern electronics.