Related papers: Well-posedness and scattering for fourth order non…

For $n\geq 3$, we study the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equations, for which the existence of the scattering operators and the global well-posedness of solutions with small data in Besov spaces…

In the present paper, we consider the Cauchy problem of nonlinear Schr\"odinger equations with a derivative nonlinearity which depends only on $\bar{u}$. The well-posedness of the equation at the scaling subcritical regularity was proved by…

We consider the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where…

In this paper we study the Cauchy problem for the elliptic and non-elliptic derivative nonlinear Schr\"odinger equations in higher spatial dimensions ($n\geq 2$) and some global well-posedness results with small initial data in critical…

We consider the cubic nonlinear fourth-order Schr\"odinger equation \[ i\partial_t u - \Delta^2 u + \mu \Delta u = \pm |u|^2 u, \quad \mu \geq 0 \] on $\mathbb{R}^N, N \geq 5$ with random initial data. We prove almost sure local…

In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schr\"odinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. The local existence of…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation with double nonlinearities with opposite sign, with one term is mass-critical and the other term is mass-supercritical and energy-subcritical, which includes the famous…

We consider the Cauchy problem of a system of quadratic derivative nonlinear Schr\"odinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. For the nonperiodic case, the author proved…

We consider the Cauchy problem for the defocusing cubic nonlinear Schr\"odinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in…

We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schr\"odinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of…

We study the wellposedness of Cauchy problem for the fourth order nonlinear Schr\"odinger equations i\partial_t u=-\eps\Delta u+\Delta^2 u+P((\partial_x^\alpha u)_{\abs{\alpha}\ls 2}, (\partial_x^\alpha \bar{u})_{\abs{\alpha}\ls 2}),\quad…

We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first…

We prove two new results about the Cauchy problem for nonlinear Schroedinger equations on four-dimensional compact manifolds. The first one concerns global wellposedness for Hartree-type nonlinearities and includes approximations of cubic…

For $n\geq 2$, we establish the smooth effects for the solutions of the linear fourth order Shr\"{o}dinger equation in anisotropic Lebesgue spaces with $\Box_k$-decomposition. Using these estimates, we study the Cauchy problem for the…

We consider the Cauchy problem for a family of semilinear defocusing Schr\"odinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle…

We prove small data scattering for the fourth-order Schr\"odinger equation with quadratic nonlinearity \begin{equation*} i\partial_t u+\Delta^2 u+\alpha u^2 + \beta \bar{u}^2=0\qquad\text{in }\mathbb{R}^5 \end{equation*} for $\alpha, \beta…

We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schr\"odinger equation $iu_t + \Delta u = |u|^{4/n} u$ for large spherically symmetric $L^2_x(\R^n)$ initial data in…

We consider the Cauchy problem for a quadratic derivative nonlinear Schr\"odinger equation whose nonlinearity is a linear combination of $\partial_x (u^2)$ and $\partial_x (|u|^2)$. We prove the local well-posedness in the $L^2$-based…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means…

In this paper, we consider the fourth-order Schr\"odinger equations with focusing, $L^2$-supercritical nonlinearity in one dimension. We prove the global existence and scattering of solutions below the ground state threshold under the…