相关论文: Global well-posedness for a NLS-KdV system on $\ma…

The Cauchy problem for a coupled system of the Schroedinger and the KdV equation is shown to be globally well-posed for data with infinite energy. The proof uses refined bilinear Strichartz estimates and the I-method introduced by…

We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\"odinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in…

In this note we prove global well-posedness for the defocusing, cubic nonlinear Schr{\"o}dinger equation with initial data lying in a critical Sobolev space.

In this article, we study the low-regularity Cauchy problem of a one dimensional quadratic Schrodinger system with coupled parameter $\alpha\in (0, 1)$. When $\frac{1}{2}<\alpha<1$,we prove the global well-posedness in $H^s(\mathbb{R})$…

We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0}\in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times…

We consider the defocusing periodic fractional nonlinear Schr\"odinger equation $$ i \partial_t u +\left(-\Delta\right)^{\alpha}u=-\lvert u \rvert ^2 u, $$ where $\frac{1}{2}< \alpha < 1$ and the operator $(-\Delta)^\alpha$ is the…

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based…

The Cauchy problem for the Zakharov system in the energy-critical dimension $d=4$ is considered. We prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground…

In this paper, we first prove global well-posedness for the defocusing cubic nonlinear Schr\"odinger equation (NLS) on 4-dimensional tori - either rational or irrational - and with initial data in $H^1$. Furthermore, we prove that if a…

We study the Cauchy problem for fractional Schr\"odinger equation with cubic convolution nonlinearity ($i\partial_t u - (-\Delta)^{\frac{\alpha}{2}}u\pm (K\ast |u|^2) u =0$) with Cauchy data in the modulation spaces $M^{p,q}(\mathbb…

We prove global well-posedness for the cubic nonlinear Schr\"odinger equation for periodic initial data in the mass-critical dimension $d=2$ for initial data of arbitrary size in the defocusing case and data below the ground state threshold…

We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d \ge 5$ with initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0} \in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d)…

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…

This paper is concerned with the Cauchy problem of the $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space $H^s({\mathbb{R}}^2)$ for $s > -1/4$, and these are optimal…

We prove the local well-posedness for the generalized Korteweg-de Vries equation in $H^s(\mathbb{R})$, $s>1/2$, under general assumptions on the nonlinearity $f(x)$, on the background of an $L^\infty_{t,x}$-function $\Psi(t,x)$, with…

We consider the initial-value problem for the Chern-Simons-Schr\"odinger system, which is a gauge-covariant Schr\"{o}dinger system in $\mathbb{R}_t\times\mathbb{R}^2_x$ with a long-range electromagnetic field. We show that, in the Coulomb…

We prove the global well-posedness for a $L^2$-critical defocusing cubic higher-order Schr\"odinger equation, namely \[ i\partial_t u + \Lambda^k u = -|u|^2 u, \] where $\Lambda=\sqrt{-\Delta}$ and $k\geq 3, k \in \mathbb{Z}$ in…

We consider the following $p$ order nonlinear half wave Schr{\"o}dinger equations$$\left(i \partial\_{t}+\partial\_{x }^2-\left|D\_{y}\right|\right) u=\pm|u|^{p-1} u$$on the plane $\mathbb{R}^2$ with $1<p\leq 2$. This equation is considered…

We consider the initial value problem (IVP) associated to a higher order nonlinear Schr\"odinger (h-NLS) equation $ \partial_{t}u+ia \partial^{2}_{x}u+ b\partial^{3}_{x}u+ic_1|u|^{2}u+c_2 |u|^{2}\partial_{x}u=0, \quad x,t \in \mathbb{R}, $…

We prove global well-posedness of the subcritical generalized Korteweg-de Vries equation (the mKdV and the gKdV with quartic power of nonlinearity) subject to an additive random perturbation. More precisely, we prove that if the driving…