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We investigate the existence and local uniqueness of normalized $k$-peak solutions for the fractional Schr\"odinger equations with attractive interactions with a class of degenerated trapping potential with non-isolated critical points.…
This is the first part of a two-paper series studying nonlinear Schr\"odinger equations with quasi-periodic initial data. In this paper, we consider the standard nonlinear Schr\"odinger equation. Under the assumption that the Fourier…
This paper investigates the existence of normalized solutions for the following Chern-Simons-Schr\"odinger equation: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\left(\frac{h^{2}(\vert x\vert)}{\vert…
In this paper we deal with the cubic Schr\"odinger system $ -\Delta u_i = \sum_{j=1}^n \beta_{ij}u_j^2 u_i$, $u_1,\dots,u_n \geq 0$ in $\mathbb{R}^N (N\leq 3)$, where $\beta=(\beta_{i,j})_{ij}$ is a symmetric matrix with real coefficients…
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…
In this paper, we study important Schr\"{o}dinger systems with linear and nonlinear couplings \begin{equation}\label{eq:diricichlet} \begin{cases} -\Delta u_1-\lambda_1 u_1=\mu_1 |u_1|^{p_1-2}u_1+r_1\beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+\kappa…
In this paper we consider a two component system of coupled non linear Schr\"odinger equations modeling the phase separation in the binary mixture of Bose-Einstein condensates and other related problems. Assuming the existence of solutions…
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…
In this article, we study the two dimensional focusing finitely and infinitely coupled cubic nonlinear Schr\"odinger system when the mass is equal to the scattering threshold. For the focusing finitely coupled cubic nonlinear Schr\"odinger…
In this paper, we report a more general class of nondegenerate soliton solutions, associated with two distinct wave numbers in different modes, for a certain class of physically important integrable two component nonlinear Schr\"{o}dinger…
This paper is concerned with a nonlinear fractional Sch\"ordinger system in $\mathbb{{R}}$ with intraspecies interactions $a_{i}>0 \ (i=1,2)$ and interspecies interactions $\beta \in\mathbb{{R}}$. We study this system by solving an…
We find a normalized solution $u=(u_1,\ldots,u_K)$ to the system of $K$ coupled nonlinear Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{l} -\Delta u_i+ \lambda_i u_i = \sum_{j=1}^K\beta_{i,j}u_i|u_i|^{p/2-2}|u_j|^{p/2}…
On a smooth, closed Riemannian manifold, we study the question of proportionality of components, also called synchronization, of vector-valued solutions to nonlinear elliptic Schr\"odinger systems with constant coefficients. In particular,…
In this paper, we study normalized solutions for the following critical Schr\"odinger-Bopp-Podolsky system: $$-\Delta u + q(x)\phi u = \lambda u + |u|^{p-2}u + \bigl(I_\alpha * |u|^{3+\alpha}\bigr)|u|^{1+\alpha}u,\quad \text{in }…
In this paper, we study the following nonlinear Schr\"{o}dinger system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -\Delta u+V(x)u=\partial_v H(x,u,v)+\omega v, \ x \in \mathbb{R}^N, \\ -\Delta v+V(x)v=\partial_u…
A nonlinear generalisation of Schrodinger's equation had previously been obtained using information-theoretic arguments. The nonlinearities in that equation were of a nonpolynomial form, equivalent to the occurence of higher-derivative…
In this paper we consider the following quasilinear Schr\"odinger-Poisson system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll} - \Delta u +\phi u = f(u) &\ \mbox{in } \Omega, -\Delta \phi - \varepsilon^{4}\Delta_4…
The paper is concerned with a nonlinear system of two coupled fractional Schr\"odinger equations with both attractive intraspecies and attractive interspecies interactions in $\mathbb{R}$. By analyzing an associated $L^2$-constrained…
We present new approaches for solving constrained multicomponent nonlinear Schr\"odinger equations in arbitrary dimensions. The idea is to introduce an artificial time and solve an extended damped second order dynamic system whose…
We prove that, for any real $\lambda$, the system $-\Delta u +\lambda u = u^3-\beta uv^2$, $ -\Delta v+\lambda v =v^3-\beta vu^2$, $ u,v\in H^1_0(\Omega),$ where $\Omega$ is a bounded smooth domain of $R^3$, admits a bounded family of…