Related papers: Sharp global well-posedness for a higher order Sch…

We prove global well-posedness for low regularity data for the $L^2-critical$ defocusing nonlinear Schr\"odinger equation (NLS) in 2d. More precisely we show that a global solution exists for initial data in the Sobolev space $H^{s}(\mathbb…

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schr\"odinger equations for the spatial dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin (2004). The first…

We study in this paper the well-posedness and stability for two linear Schr\"odinger equations in $d$-dimensional open bounded domain under Dirichlet boundary conditions with an infinite memory. First, we establish the well-posedness in the…

In this paper, we study the probabilistic local well-posedness of the cubic Schr\"odinger equation (cubic NLS): \[ (i\partial_{t} + \Delta) u = \pm |u|^{2} u \text{ on } [0,T) \times \mathbb{R}^{d}, \] with initial data being a Wiener…

We prove global well-posedness for the Cauchy problem associated with the Kadomotsev-Petviashvili-Burgers equation (KPBII) in $\mathbb R^2$ when the initial value belongs to the anisotropic Sobolev space $H^{s_1,s_2}(\mathbb R^2)$ for all…

In [12], we proved that $1$-d periodic fractional Schr\"odinger equation with cubic nonlinearity is locally well-posed in $H^s$ for $s>\frac{1-\alpha}{2}$ and globally well-posed for $s>\frac{5\alpha-1}{6}$. In this paper we define an…

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…

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schr\"odinger equations. This system was introduced by M. Colin and T. Colin (2004). The first and second authors obtained some…

We consider the initial value problem (IVP) associated to the Schr\"odinger-Debye system posed on a $d$-dimensional compact Riemannian manifold $M$ and prove local well-posedness result for given data $(u_0, v_0)\in H^s(M)\times (H^s(M)\cap…

We study the Cauchy problem for the following Majda-Biello system in the case $\alpha=4$, where the resonance effect is the most significant, on the real line. \[ \left\{ \begin{array}{rcl} u_{t} + u_{xxx} & = & - v v_x, v_{t} + \alpha…

We prove that the critical Maxwell-Klein Gordon equation on R4+1 is globally well-posed for smooth initial data which are small in the energy. This reduces the problem of global regularity for large, smooth initial data to precluding…

We consider the inhomogeneous biharmonic nonlinear Schr\"odinger equation $$ i u_t +\Delta^2 u+\lambda|x|^{-b}|u|^\alpha u = 0, $$ where $\lambda=\pm 1$ and $\alpha$, $b>0$. In the subctritical case, we improve the global well-posedness…

We study the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation. We prove that results about local and global well-posedness for initial data in $H^s(R)$, with $s>-1/2$, are sharp in the…

Local and global well-posedness results are established for the initial value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving Ill-posedness results otherwise. The…

We consider the nonlinear Schr\"odinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced…

We consider the Cauchy problem of a class of higher order Schr\"odinger type equations with constant coefficients. By employing the energy inequality, we show the $L^2$ well-posedness, the parabolic smoothing and a breakdown of the…

This paper discusses the initial-boundary-value problems (IBVP) of nonlinear Schr\"odinger equations posed in a half plane $\mathbb{R} \times \mathbb{R}^+$ with nonhomogeneous Dirichlet boundary conditions. For any given $s \ge 0$, if the…

In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u-V(x)u…

This work is concerned with the Cauchy problem for a coupled Schr\"odinger-Benjamin-Ono system $$\left \{ \begin{array}{l} i\partial_tu+\partial_x^2u=\alpha uv,\qquad t\!\in\![-T,T], \ x\!\in\!\mathbb R,\\ \partial_tv+\nu\mathcal…

This paper is devoted to the well-posedness of stochastic nonlinear Schr\"odinger equations in the energy space H1(Rd), which is a natural continuation of our recent work [1]. We consider both focusing and defocusing nonlinearities and…