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About simple nonlinear and linear superpositions of special exact solutions of Veselov-Novikov equation. (English) Zbl 1298.37049
Summary: New exact solutions, nonstationary and stationary, of Veselov-Novikov (VN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number $$N$$ of exact special solutions $$u^{(n)},n=1,\dots, N$$ are constructed via Zakharov and Manakov $$\overline{\partial }$$-dressing method. Simple nonlinear superpositions are represented up to a constant by the sums of solutions $$u^{(n)}$$ and calculated by $$\overline{\partial }$$-dressing on nonzero energy level of the first auxiliary linear problem, i.e., 2D stationary Schrödinger equation. It is remarkable that in the zero energy limit simple nonlinear superpositions convert to linear ones in the form of the sums of special solutions $$u^{(n)}$$. It is shown that the sums $$u= u^{(k_1)}+\ldots + u^{(k_m)}$$, $$1 \leq k{_1} < k{_2} < \dots < k{_m} \leq N$$ of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schrödinger equation and can serve as model potentials for electrons in planar structures of modern electronics.{