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