Related papers: Modified scattering for the critical nonlinear Sch…

In this paper, we consider the Cauchy's problem of global existence and scattering behavior of small, smooth, and localized solutions of cubic fractional Schr\"odinger equations in one dimension, \begin{equation*} \mathrm{i} \partial_t u-…

We study the initial value problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with symmetric boundary conditions: $q(x,t)\to Ae^{2iA^2t}$ as $x\to\pm\infty$,…

In this paper, we consider existence of positive solutions for the Schr\"odinger quasilinear elliptic problem $$ \left\{ \begin{array}{l} \Delta_pu+\Delta_p(|u|^{2\gamma})|u|^{2\gamma-2}u = a(x)g(u)~ \mbox{on}~ \mathbb{R}^N,\\ u>0\…

We prove existence of normalized solutions to \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 & \text{in $\mathbb{R}^3$} -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v & \text{in $\mathbb{R}^3$}\int_{\mathbb{R}^3} u^2 =…

We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - \Delta u + u \pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in the…

We consider the scattering results of the radial solutions below the ground state to the focusing inhomogeneous nonlinear Schr\"odinger equation $$i\partial_tu+\Delta u +|x|^{-b}|u|^{p}u=0$$ in two dimension, where $0<b<1$ and…

We present numerical simulations of the defocusing nonlinear Schrodinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig-Merle and Kilip-Visan who considered some energy…

In this paper, we consider the singularly perturbed fractional Schr\"{o}dinger equation \begin{equation*} \epsilon^{2\alpha}(-\Delta)^\alpha u+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $\epsilon>0$ is a small parameter,…

We consider the cubic nonlinear Schr\"{o}dinger equation on the star graph with the Kirchhoff boundary condition. We prove modified scattering for the final state problem and the initial value problem. Moreover, we also consider the failure…

In this paper we consider Schr{\"o}dinger equations with nonlinearities of odd order 2$\sigma$ + 1 on T^d. We prove that for $\sigma$d$\ge$2, they are strongly illposed in the Sobolev space H^s for any s \textless{} 0, exhibiting…

The aim of this work is the study of the existence of normalized solutions to the nonlinear Schr\"odinger equation with nonlocal nonlinearities: \begin{equation}\nonumber \left\{\begin{aligned} &-\Delta u =\lambda…

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 consider the following class of focusing $L^2$-supercritical fourth-order nonlinear Schr\"odinger equations \[ i\partial_t u - \Delta^2 u + \mu \Delta u = - |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N, \] where $N\geq…

We consider non-gauge-invariant cubic nonlinear Schr\"odinger equations in one space dimension. We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^\infty$ decay up to time…

We mainly consider the focusing biharmonic Schr\"odinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(\Delta^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{N}}, u(0,…

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^\alpha u = 0\quad\text{on}\quad\mathbb{R}\times\mathbb{R}^N, \] with $N\geq 2$, $0<b<\min\{\tfrac{N}{2},2\}$, and…

In this paper, we consider the following nonlinear Schr\"{o}dinger equations with mixed nonlinearities: \begin{eqnarray*} \left\{\aligned &-\Delta u=\lambda u+\mu |u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ &u\in…

In this article, we study the existence of normalized ground state solutions for the following biharmonic nonlinear Schr\"{o}dinger equation with combined nonlinearities \begin{equation*} \Delta^2u=\lambda u+\mu|u|^{q-2}u+|u|^{p-2}u,\quad…

We prove scattering below the mass-energy threshold for the focusing inhomogeneous nonlinear Schr\"odinger equation \begin{equation} iu_t + \Delta u + |x|^{-b}|u|^{p-1}u=0, \end{equation} when $b \geq 0$ and $N > 2$ in the intercritical…

Using a modified version of Schauder's fixed point theorem, measures of non-compactness and classical techniques, we provide new general results on the asymptotic behavior and the non-oscillation of second order scalar nonlinear…