Novikov-Veselov equation is a (2+1)-dimensional analog of the classic Korteweg-de Vries equation integrable via the inverse scattering translform for the 2-dimensional stationary Schrödinger equation. In this talk we present some recent results on existence and absence of algebraically localized solitons for the Novikov-Veselov equation as well as some results on the large time behavior of the “inverse scattering solutions” for this equation.

Two new applications of ${\Lambda }^2$-representations of PDEs are presented: 1. Geometric algorithms for numerical integration of PDEs by constructing planimetric discrete nets on the Lobachevsky plane ${\Lambda }^2$. 2. Employing ${\Lambda }^2$-representations for the spectral-evolutionary problem for nonlinear PDEs within the inverse scattering problem method.