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We consider the Cauchy problem for the two-dimensional Novikov-Veselovequation integrable via the inverse scattering problem for the Schr\"odingeroperator with fixed negative energy. The associated linear equation ischaracterized by a rational symbol which is not a polynomial, except when theenergy parameter is zero. With the help of a complex analysis point of view ofthe problem, we establish uniform decay estimates for the linear solution withgain of almost one derivative, and we use this result together with Fourierdecomposition methods and $X^{s,b}$ spaces to prove local well-posedness in$H^s$, $s>\frac12$.

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