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In the present paper we are concerned with the Novikov--Veselov equation atnegative energy, i.e. with the $ (2 + 1) $--dimensional analog of the KdVequation integrable by the method of inverse scattering for thetwo--dimensional Schr\"odinger equation at negative energy. We show that thesolution of the Cauchy problem for this equation with non--singular scatteringdata behaves asymptotically as $ \frac{\const}{t^{3/4}} $ in the uniform normat large times $ t $. We also present some arguments which indicate that thisasymptotics is optimal.

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