New lumps of Veselov-Novicov equation and new exact rational potentials of Schroedinger equation with multiple pole wave functions via $\overline\partial$-dressing method

Abstract:
The scheme for calculating via Zakharov-Manakov $\overline\partial$-dressing method of new rational solutions with constant asymptotic values at infinity of the famous two-dimensional Veselov-Novikov (VN) integrable nonlinear evolution equation and new exact rational potentials of two-dimensional stationary Schroedinger (2DSchr) equation with multiple pole wave functions is developed. As examples new lumps of VN nonlinear equation and new exact rational potentials of 2DSchr equation with multiple pole of order two wave functions are calculated. Among the constructed rational solutions are as nonsingular and also singular.