Related papers: An averaging theorem for nonlinear Schr\"odinger e…

Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$ u_t+i(-\Delta u+V(x)u)=\epsilon\mu\Delta u+\epsilon \mathcal{P}( u),\quad x\in {R^d}\,, \quad(*) $$ under the periodic boundary conditions, where…

We consider the free linear Schr\"odinger equation on a torus $\mathbb T^d$, perturbed by a hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$ u_t -i\Delta u +i\nu \rho |u|^{2q_*}u = - \nu f(-\Delta) u +…

We consider the free linear Schroedinger equation on a torus $\mathbb T^d$, perturbed by a Hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$u_t -i\Delta u +i\nu \rho |u|^{2q_*}u = - \nu f(-\Delta) u +…

We consider the linear Schr\"odinger equation under periodic boundary condition, driven by a random force and damped by a quasilinear damping: $$ \frac{d}{dt}u+i\big(-\Delta+V(x)\big) u=\nu \Big(\Delta u-\gr |u|^{2p}u-i\gi |u|^{2q}u \Big)…

We consider a number of linear and non-linear boundary value problems involving generalized Schr\"odinger equations. The model case is $-\Delta u=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. We use the Sobolev…

This paper deals with the 2-D Schr\"odinger equation with time-oscillating exponential nonlinearity $i\partial_t u+\Delta u= \theta(\omega t)\big(e^{4\pi|u|^2}-1\big)$, where $\theta$ is a periodic $C^1$-function. We prove that for a class…

Consider a weakly nonlinear CGL equation on the torus~$\mathbb{T}^d$: \[u_t+i\Delta u=\epsilon [\mu(-1)^{m-1}\Delta^{m} u+b|u|^{2p}u+ ic|u|^{2q}u].\eqno{(*)}\] Here $u=u(t,x)$, $x\in\mathbb{T}^d$, $0<\epsilon<<1$, $\mu\geqslant0$,…

We consider the one-dimensional nonlinear Schr\"odinger equation $$ iu_t + u_{xx} + \mathcal{N}(u)u=0, \quad x,t \in \mathbb R, $$ with the nonlinearity term that is expressed as a sum of powers, possibly infinite: $$ \mathcal{N}(u) = \sum…

We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…

We establish that the quadratic non-linear Schr\"odinger equation $$ iu_t + u_{xx} = u^2$$ where $u: \R \times \R \to \C$, is locally well-posed in $H^s(\R)$ when $s \geq -1$ and ill-posed when $s < -1$. Previous work of Kenig, Ponce and…

In this paper, we study the long-time behavior for the mass-critical nonlinear Schr\"odinger equation on the line \[ i\partial_t u + \partial_x^2 u = |u|^4 u, u(0, x) = u_0 \in L_x^2(\Bbb R). \] The global well-posedness and scattering for…

In this paper, we consider the following weakly coupled nonlinear Schr\"odinger system \begin{equation*} \left\{ \begin{array}{ll} -\epsilon^{2}\Delta u_1 + V_1(x)u_1 = |u_1|^{2p - 2}u_1 + \beta|u_1|^{p - 2}|u_2|^pu_1, & x\in…

In this paper we study the following class of fractional relativistic Schr\"odinger equations: \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in…

We prove a Nekhoroshev type theorem for the nonlinear Schr\"odinger equation $$ iu_t=-\Delta u+V\star u+\partial_{\bar u}g(u,\bar u)\, \quad x\in \T^d, $$ where $V$ is a typical smooth potential and $g$ is analytic in both variables. More…

We consider the $d$-dimensional nonlinear Schr\"odinger equation under periodic boundary conditions: $-i\dot u=-\Delta u+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u), \quad u=u(t,x), x\in\T^d $ where $V(x)=\sum \hat V(a)e^{i\sc{a,x}}$ is…

In this paper, we study the following semilinear Schr\"odinger equation $$ -\epsilon^2\triangle u+ u+ V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}), $$ where $N\geq 2$ and $\epsilon>0$ is a small parameter. The function $V$ is bounded in…

We consider the nonlinear Schr\"{o}dinger equation $-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-\Delta+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in…

We study the Schr\"odinger equations $-\Delta u + V(x)u = f(x,u)$ in $\mathbb{R}^N$ and $-\Delta u - \lambda u = f(x,u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$. We assume that $f$ is superlinear but of subcritical growth and…

In this paper, we consider a derivative nonlinear Schr\"odinger equation $$ \mathrm{i}\partial_{t}u+\partial_{xx}u-V\ast u+\mathrm{i}\vert u\vert^{2}\partial_{x}u=0 $$ on the torus $\mathbb{T}$, depending on some potential $V$. We prove…

We consider the nonlinear Schr\"odinger equation $iu_t + \Delta u= \lambda |u|^{\frac {2} {N}} u $ in all dimensions $N\ge 1$, where $\lambda \in {\mathbb C}$ and $\Im \lambda \le 0$. We construct a class of initial values for which the…