New Lumps of Veselov–Novikov Integrable Nonlinear Equation and New Exact Rational Potentials of Two-Dimensional Stationary Schrödinger Equation via ∂-Dressing Method

Exact solutions of differential equations of physics are very important for the understanding of various physical phenomena. The generation and application of new methods of calculation of exact solutions was and is actual task in all times for human scientific civilization. In the last two decades the Inverse Spectral Transform (IST) method has been generalized and successfully applied to various two-dimensional nonlinear evolution equations such as Kadomtsev–Petvashvili, Davey–Stewartson, Nizhnik–Veselov–Novikov, Zakharov–Manakov system, Ishimori, two-dimensional integrable sine-Gordon and others (see books [1–4] and references therein). The nonlocal Riemann–Hilbert problem [5], ∂-problem [6] and more general ∂-dressing method of Zakharov and Manakov [7–10] are now basic tools for solving two-dimensional integrable nonlinear evolution equations. Great task for mathematicians and physicists now is the generalization of IST method to multidimensional differential equations of mathematical physics. In the present short note the results of recent calculations [19] of new exact rational so called multiple pole solutions of the famous two-dimensional integrable Veselov–Novikov (VN) nonlinear equation [11]