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Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity (1505.06496v1)
Published 24 May 2015 in math.AP
Abstract: In the present paper, we consider the Cauchy problem of fourth order nonlinear Schr\"odinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schr\"odinger equation with the derivative quartic nonlinearity $\partial _x (\overline{u}4)$ is the small data global in time well-posed and scattering to a free solution. Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (um)$ with $(m-1)d \ge 4$.