Related papers: On the existence of vector solutions to nonlinear …
We consider the standing-wave problem for a nonlinear Schr\"{o}dinger equation, corresponding to the semilinear elliptic problem \begin{equation*} -\Delta u+V(x)u=|u|^{p-1}u,\ u\in H^1(\mathbb{R}^2), \end{equation*} where $V(x)$ is a…
We derive an effective equation for the dynamics of many identical bosons in dimension one in the presence of a tiny impurity. The interaction between every pair of bosons is mediated by the impurity through a positive three-body potential.…
We consider a system of coupled nonlinear Schr{\"o}dinger equations in one space dimension. First, we prove the existence of multi-speed solitary waves, i.e solutions to the system with each component behaving at large times as a solitary…
We investigate rogue-wave solutions in a three-component coupled nonlinear Schrodinger equation. With certain requirements on the backgrounds of components, we construct a multi-rogue-wave solution that exhibits a structure like a…
We consider the one-dimensional nonlinear Schr\"odinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family of solitary wave solutions in both the focusing and…
In this paper, we study the following Schr\"{o}dinger-Born-infeld system with a general nonlinearity $$ \left\{ \begin{array}{ll} -\triangle u+u+\phi u=f(u)+\mu|u|^4u\,\,&\mbox{in}\,\,\R^3,\\…
We prove the existence of a new type of solutions to a nonlinear Schr\"odinger system. These solutions, which we call "multi-speeds solitary waves", are behaving at large time as a couple of scalar solitary waves traveling at different…
We present a new method for the solution of the Schrodinger equation applicable to problems of non-perturbative nature. The method works by identifying three different scales in the problem, which then are treated independently: An…
We study the following nonlinear Schr\"{o}dinger system which is related to Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1 u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in \Omega, -\Delta…
In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation \begin{equation*} -\Delta w+ V\left( \left| x\right| \right) w - w \left( \Delta w^2 \right)= K(|x|) g(w) \quad \text{in…
In this paper we study the coupled Schr\"odinger-Maxwell system $$\left\{ \begin{array}{lll} -\triangle u+u +\phi u=\lambda \alpha(x) f(u)& {\rm in} & \mathbb R^3,\\ -\triangle \phi =u^2 & {\rm in} & \mathbb R^3, \end{array}\right. $$ where…
We study a class of $3D$ quadratic Schr\"odinger equations as follows, $(\partial_t -i \Delta) u = Q(u, \bar{u})$. Different from nonlinearities of the $uu$ type and the $\bar{u}\bar{u}$ type, which have been studied by…
In the 2-d setting, given an $H^1$ solution $v(t)$ to the linear Schr\"odinger equation $i\partial_t v +\Delta v =0$, we prove the existence (but not uniqueness) of an $H^1$ solution $u(t)$ to the defocusing nonlinear Schr\"odinger (NLS)…
We treat the stationary (cubic) nonlinear Schr\"odinger equation (NSLE) on simplest graphs. Formulation of the problem and exact analytical solutions of NLSE are presented for star graphs consisting of three bonds. It is shown that the…
We construct solutions to the nonlinear magnetic Schr\"odinger equation $$ \left\{ \begin{aligned} - \varepsilon^2 \Delta_{A/\varepsilon^2} u + V u &= \lvert u\rvert^{p-2} u & &\text{in}\ \Omega,\\ u &= 0 & &\text{on}\ \partial\Omega,…
We study positive bound states for the semiclassical stationary nonlinear Schr\"odinger equation. We are especially interested in solutions which concentrate on a lower dimensional sphere. We adopt a purely variational approach which allows…
For the first time, Schr\"odinger equations with cubic and more complex nonlinearities containing the unknown function with constant delay are analyzed. The physical considerations that can lead to the appearance of a delay in such…
We use a fractional transformation to connect the traveling wave solutions of the nonlinear Schr\"odinger equation (NLSE), phase-locked with a source, to the elliptic functions satisfying, $f^{\prime\prime}\pm af\pm \lambda f^{3}=0$. The…
We consider the long time behavior of the solutions of the coupled Schr\"odinger-KdV systems \begin{eqnarray*} \left\{ \begin{array}{llll}i\partial_tu+\partial^2_xu=\alpha uv+\beta u|u|^2,\hskip30pt (x,t)\in \mathbb{R}\times…
We study the following zero-mass Schr{\"o}dinger-Poisson-Slater equation \[ - \Delta u + \left( \frac{1}{4 \pi | x |} \ast u^2 \right) u = f (| x |, u) \text{,} \qquad u \in \mathcal{D}^{1, 2} (\mathbb{R}^3) \text{} \] with nonlinearity…