Related papers: Wellposedness of NLS in Modulation Spaces
We consider the dispersion managed nonlinear Schr\"dinger equations with quintic and cubic nonlinearities in one and two dimensions, respectively. We prove the global well-posedness and scattering in $L_x^2$ for small initial data employing…
We prove a local in time well-posedness result for quasi-linear Hamiltonian Schr\"odinger equations on $\mathbb{T}^d$ for any $d\geq 1$. For any initial condition in the Sobolev space $H^s$, with $s$ large, we prove the existence and…
We prove global well-posedness and scattering for the nonlinear Schr\"odinger equation with power-type nonlinearity \begin{equation*} \begin{cases} i u_t +\Delta u = |u|^p u, \quad \frac{4}{n}<p<\frac{4}{n-2}, u(0,x) = u_0(x)\in H^s(\R^n),…
We study the Cauchy problem for a generalized derivative nonlinear Schr\"odinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions are constructed…
In this paper, we consider the cubic fourth-order nonlinear Schr\"odinger equation (4NLS) under the periodic boundary condition. We prove two results. One is the local well-posedness in $H^s$ with $-1/3 \le s < 0$ for the Cauchy problem of…
In this paper, we study cubic and quintic nonlinear Schr\"odinger systems on 3-dimensional tori, with initial data in an adapted Hilbert space $H^s_{\underline{\lambda}},$ and all of our results hold on rational and irrational rectangular,…
We consider the Cauchy problem of the cubic nonlinear Schr\"odinger equation (NLS) on $\mathbb R^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling critical regularity $s_\text{crit} =…
In this paper, we prove the global well-posedness of the energy-critical nonlinear Schr\"odinger equations on the torus $\mathbb{T}^{d}$ for general dimensions. This result is new for dimensions $d\ge5$, extending previous results for…
We address stability of multi-solitons in the cubic NLS (nonlinear Schr\"{o}dinger) equation on the line. By using the dressing transformation and the inverse scattering transform methods, we obtain the orbital stability of multi-solitons…
By applying a simple symmetry reduction on a two-layer liquid model, a nonlocal counterpart of it is obtained. Then a general form of nonlocal nonlinear Schrodinger (NNLS) equation with shifted parity, charge-conjugate and delayed time…
We consider NLS on $\T^2$ with multiplicative spatial white noise and nonlinearity between cubic and quartic. We prove global existence, uniqueness and convergence almost surely of solutions to a family of properly regularized and…
In this paper, we show that the one dimensional septic nonlinear Schr\"odinger equation is globally well-posed and scatters in $H^s (\mathbb{R})$ when $s > 19/54$. We prove new smoothing estimates on the nonlinear Duhamel part of the…
In this paper, we consider the nonlinear Schr\"odinger equation $iu_t +\Delta u= \lambda |u|^{\frac {4} {N-4}} u$ in $\R^N $, $N\ge 5$, with $\lambda \in \C$. We prove local well-posedness (local existence, unconditional uniqueness,…
We consider the initial value problem (IVP) associated to the cubic nonlinear Schr\"odinger equation with third-order dispersion \begin{equation*} \partial_{t}u+i\alpha \partial^{2}_{x}u- \partial^{3}_{x}u+i\beta|u|^{2}u = 0, \quad x,t \in…
We establish global well-posedness for both the defocusing and focusing complex-valued modified Korteweg--de Vries equations on the real line in modulation spaces $M_p^{s,2}(\mathbb{R})$, for all $1\leq p<\infty$ and $0\leq s<3/2-1/p$. We…
We use the dispersive properties of the linear Schr\"{o}dinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $\mathbb{R}^d$ for $d\geq 2$. The proofs…
We prove local and global well-posedness results for the Gabitov-Turitsyn or dispersion managed nonlinear Schr\"odinger equation with a large class of nonlinearities and arbitrary average dispersion on $L^2(\mathbb{R})$ and…
We will show its local well-posedness in modulation spaces $M^{1/2}_{2,q}({\Real})$ $(2\leq q<\infty) $. It is well-known that $H^{1/2}$ is a critical Sobolev space of DNLS so that it is locally well-posedness in $H^s$ for $s\geq 1/2$ and…
We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schr\"odinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the…
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…