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In this paper we study the Cauchy problem for the elliptic and non-elliptic derivative nonlinear Schrodinger equations in higher spatial dimensions ($ngeq 2$) and some global well-posedness results with small initial data in critical Besov spaces $B^s_{2,1}$ are obtained. As by-products, the scattering results with small initial data are also obtained.
We prove that the derivative nonlinear Schr{o}dinger equation is globally well-posed in $H^{frac 12} (mathbb{R})$ when the mass of initial data is strictly less than $4pi$.
This paper is concerned with the derivative nonlinear Schr{o}dinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small amplitude is proved.
We consider the large time behavior in two types of equations, posed on the whole space R^d: the Schr{o}dinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier-Stokes equations on
We study the inverse scattering problem for the three dimensional nonlinear Schroedinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scatt
We analyze dynamical properties of the logarithmic Schr{o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asy