Collapse in the nonlocal nonlinear Schr\"odinger equation
Reads0
Chats0
TLDR
In this article, the authors studied the effect of singular non-local kernels in arbitrary dimension using both Lyapunoff's method and virial identities, and proved that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimensions collapse does not occur.Abstract:
We discuss spatial dynamics and collapse scenarios of localized waves governed by the nonlinear Schr\"{o}dinger equation with nonlocal nonlinearity. Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimension collapse does not occur. Then we study in detail the effect of singular nonlocal kernels in arbitrary dimension using both, Lyapunoff's method and virial identities. We find that for for a one-dimensional case, i.e. for $n=1$, collapse cannot happen for nonlocal nonlinearity. On the other hand, for spatial dimension $n\geq2$ and singular kernel $\sim 1/r^\alpha$, no collapse takes place if $\alpha<2$, whereas collapse is possible if $\alpha\ge2$. Self-similar solutions allow us to find an expression for the critical distance (or time) at which collapse should occur in the particular case of $\sim 1/r^2$ kernels. Moreover, different evolution scenarios for the three dimensional physically relevant case of Bose Einstein condensate are studied numerically for both, the ground state and a higher order toroidal state with and without an additional local repulsive nonlinear interaction. In particular, we show that presence of an additional local repulsive term can prevent collapse in those cases.read more
Citations
More filters
Book ChapterDOI
Quantum and Nonlinear Optics in Strongly Interacting Atomic Ensembles
Callum R. Murray,Thomas Pohl +1 more
TL;DR: In this article, the authors present an overview of this rapidly developing field, from classical effects to quantum manifestations of the nonlocal nonlinearities emerging in such systems, describing the many experimental breakthroughs so far demonstrated and discuss potential applications looming on the horizon.
Book ChapterDOI
Quantum and Nonlinear Optics in Strongly Interacting Atomic Ensembles
Callum R. Murray,Thomas Pohl +1 more
TL;DR: In this paper, the authors present an overview of this rapidly developing field, from classical effects to quantum manifestations of the nonlocal nonlinearities emerging in such systems, describing the many experimental breakthroughs so far demonstrated and discuss potential applications looming on the horizon.
Journal ArticleDOI
Quasiperiodic oscillations and homoclinic orbits in the nonlinear nonlocal Schrödinger equation
TL;DR: In this article, a linear stability analysis of higher-order bright solitons is performed by solving the Bogoliubov-de Gennes equations, which enables us to understand the emergence of a new oscillatory state as a growing unstable mode of a higherorder soliton.
Journal ArticleDOI
Stability of solitary waves in random nonlocal nonlinear media
TL;DR: In this article, the authors consider the interplay between nonlocal nonlinearity and randomness for two different nonlinear Schrodinger models and show that the stability of bright solitons in the presence of random perturbations increases dramatically with the correlation length of the noise in the transverse plane.
Journal ArticleDOI
On dyadic nonlocal Schrödinger equations with Besov initial data
TL;DR: In this article, the authors considered the pointwise convergence of the Schrodinger-Dirac equation to the initial data in a dyadic Besov space and provided a summability formula for the kernel of the fractional derivative of order β associated to the dyadic distance on R +.
References
More filters
Journal ArticleDOI
Many-Body Physics with Ultracold Gases
TL;DR: In this article, a review of recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases is presented, focusing on effects beyond standard weakcoupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation.
Journal ArticleDOI
Theory of Bose-Einstein condensation in trapped gases
TL;DR: In this article, the authors reviewed the Bose-Einstein condensation of dilute gases in traps from a theoretical perspective and provided a framework to understand the main features of the condensation and role of interactions between particles.
Book
Optical solitons : from fibers to photonic crystals
TL;DR: In this article, the authors introduce spatial and temporal solitons in photonic crystals, and introduce the concept of Incoherent Solitons, which is a subclass of the spatial and temporally soliton.
Journal ArticleDOI
Theory of ultracold atomic Fermi gases
TL;DR: In this article, the physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective, focusing on the effect of interactions that bring the gas into a superfluid phase at low temperature.
BookDOI
The nonlinear Schrödinger equation : self-focusing and wave collapse
TL;DR: In this article, the authors present a basic framework to understand structural properties and long-time behavior of standing wave solutions and their relationship to a mean field generation and acoustic wave coupling.