Related papers: Modular Schr\"{o}dinger equation and dynamical dua…
The dynamics of models described by a one-dimensional discrete nonlinear Schr\"odinger equation is studied. The nonlinearity in these models appears due to the coupling of the electronic motion to optical oscillators which are treated in…
We examine a recently-proposed family of nonlinear Schr\"odinger equations [J. Phys. A: Math. Gen. 27:1771(1994)] with respect to a group of transformations that linearize a subfamily of them. We investigate the structure of the whole…
We consider a system of two discrete nonlinear Schr\"{o}dinger equations, coupled by nonlinear and linear terms. For various physically relevant cases, we derive a modulational instability criterion for plane-wave solutions. We also find…
In this paper we deal with a nonlinear Schr\"{o}dinger equation with chaotic, random, and nonperiodic cubic nonlinearity. Our goal is to study the soliton evolution, with the strength of the nonlinearity perturbed in the space and time…
We study a 1D nonlinear Schr{\"o}dinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied…
The set of integrable symmetries of the nonstationary Schr\"{o}dinger equation is shown to admit a natural decomposition into subsets of mutually commuting symmetries. Hierarchies of time evolutions associated with each of these subsets…
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…
In this Letter we study theoretically the interaction of optical waves in nonlinear dynamical medium, i.e. medium with relaxation. Taking into account the relaxation of the photoinduced nonlinearity we derive a single evolution equation,…
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 discuss the observability of a one-dimensional Schr\"odinger equation on certain time dependent domain. In linear moving case, we give the exact boundary and pointwise internal observability for arbitrary time. For the general moving, we…
We consider a derivative nonlinear Schr\"odinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the…
We examine the modulational and parametric instabilities arising in a non-autonomous, discrete nonlinear Schr{\"o}dinger equation setting. The principal motivation for our study stems from the dynamics of Bose-Einstein condensates trapped…
In the context of quantum mechanics superoscillations, or the more general supershifts, appear as initial conditions of the time dependent Schr\"odinger equation. Already in \cite{ABCS21_2} a unified approach was developed, which yields…
Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given…
In this note, we give an overview of some results obtained in [3]. This latter work is devoted to the study of the one-dimensional nonlinear Schr{\"o}dinger equation with random initial conditions. Namely, we describe the nonlinear…
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…
Considered is the Schr\"odinger equation in a finite-dimensional space as an equation of mathematical physics derivable from the variational principle and treatable in terms of the Lagrange-Hamilton formalism. It provides an interesting…
We consider time-dependent Schroedinger equations in one dimension with double well potential and an external nonlinear perturbation. If the initial state belongs to the eigenspace spanned by the eigenvectors associated to the two lowest…
We study an inverse problem related to the dynamical Schr{\"o}dinger equation in a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linear Schr\"odinger equation possesses a trivial solution, we linearize the equation around the…
We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schr\"odinger type and have recently been obtained in \cite{DLS}…