We consider the Schr\"odinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +\Delta u=\lambda|u|^{\alpha}u \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $\lambda\in {\mathbb C} $ $\Im\lambda<0$. Assuming $\frac {2} {N+2}<\alpha<\frac2N$, we give a precise description of long-time behavior solutions (including decay rates $L^2$ and $L^\infty asymptotic profile), fo...

We introduce a tool that solves the Schr\"odinger-Euler-Poisson system of equations and allows study interaction between ultralight bosonic dark matter, whose dynamics is described with Schr\"odinger-Poisson luminous matter which, as first approximation, modeled single component compressible ideal fluid. The two fields are coupled through Poisson equation, source addition both, fluid densities....

We consider three-parameter Yang-Baxter deformations of the $AdS_5\times T^{1,1}$ superstring for abelian $r$-matrices which are solutions classical equation. find two new backgrounds dual to dipole deformed Klebanov-Witten gauge theory and nonrelativistic with Schr\"odinger symmetry

For the Schr\"odinger equation with a cubic-quintic, focusing-defocusing nonlinearity in one space dimension, we prove asymptotic stability of solitary waves for large range admissible frequencies. this model, linearized problem around does not have internal mode nor resonance.

In this paper, we start with the quantum Hamilton-Jacobi approach and show that underlying complex pole evolution of Schr\"odinger equation is described by action in terms a random matrix. The wave function given matrix probability distribution function. literature known as famous Cole-Hopf Transformation.

We study concentrated bound states of the Schrr odinger-Newton equations

We discuss a discrete approach to the multiscale reductive perturbative method and apply it biatomic chain with nonlinear interaction between atoms. This system is important describe time evolution of localized solitonic excitations. require that also reduced equation be discrete. To do so coherently we need discretize variable able get asymptotic waves carry out expansion around them. Our resu...

We prove that the spectrum of the Schrr odinger operator with periodic electric and magnetic potentials is absolutely continuous.