A large-time asymptotics for the solution of the Cauchy problem for the Novikov–Veselov equation at negative energy with non-singular scattering data

Abstract

Abstract. In the present paper we are concerned with the Novikov-Veselov equation at negative energy, i.e. with the (2 + 1)-dimensional analog of the KdV equation integrable by the method of inverse scattering for the two-dimensional Schrödinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non-singular scattering data behaves asymptotically as const t 3/4 in the uniform norm at large times t. We also prove that this asymptotics is optimal.

IST Versus PDE: A Comparative Study

“…We also refer to [99] and to the papers [100,101,102,103,104,105,177] for many interesting results on the soliton solutions (absence, decay properties,..) and the long time asymptotics. More precisely it is proven that the rational, non singular solutions introduced in [79,80] at positive energy are multi-solitons.…”

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation

“…The dispersive estimate is proved via a stationary-phase type procedure for a two-dimensional phase depending on a complex parameter. This type of procedure has been previously used in [14,16] in the framework of the Inverse Scattering Transformation (IST) approach to NV. An important role in our estimations is played by certain changes of variables, arising naturally in the IST method for NV.…”

Well-posedness and ill-posedness results for the Novikov-Veselov equation

“…There is plenty of literature around this equation through different methods (the inverse scattering method for instance), and in different formulations (at non zero energy, in our case the energy is 0). The interested reader can look at [8], [9], [10], [11], [6], [7], [12], [13], [19] for different results on a variety of problems concerning (1.1).…”