Anna Kazeykina

Large time asymptotics for the Grinevich-Zakharov potentials

Abstract In this article we show that the large time asymptotics for the Grinevich–Zakharov rational solutions of the Novikov–Veselov equation at positive energy (an analog of KdV in 2 + 1 dimensions) is given by a finite sum of localized travel waves (solitons).

A LARGE TIME ASYMPTOTICS FOR TRANSPARENT POTENTIALS FOR THE NOVIKOV–VESELOV EQUATION AT POSITIVE ENERGY

In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for the Novikov–Veselov equation (an analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are focused on a family of reflectionless (transparent) potentials parameterized by a function of two variables. In particular, we show that there are no isolated soliton type waves in the large time asymptotics for these solutions in contrast with well-known large time asymptotics for solutions of the KdV equation with reflectionless initial data.

Absence of exponentially localized solitons for the Novikov–Veselov equation at negative energy

We show that the Novikov–Veselov equation (an analogue of KdV in dimension 2 + 1) does not have exponentially localized solitons at negative energy.

Absence of solitons with sufficient algebraic localization for the Novikov-Veselov equation at nonzero energy

It is shown that the Novikov-Veselov equation (an analogue of the KdV equation in dimension 2 + 1) at positive and negative energies does not have solitons with space localization stronger than O(|x|−3) as |x| →∞.

Examples of the absence of a traveling wave for the generalized Korteweg-de Vries-Burgers equation

An example of convex function f(u) for which the generalized Korteweg-de Vries-Burgers equation ut + (f(u))x + auxxx − buxx = 0 has no solutions in the form of a traveling wave with specified limits at infinity is constructed. This example demonstrates the difficulties in analyzing asymptotic behavior of the Cauchy problem for the Korteweg-de Vries-Burgers equation that is not inherent in the type of equation for the conservation law, the Burgers-type equation, and its finite difference analog.