812 research outputs found
Nonlinear Phenomena of Ultracold Atomic Gases in Optical Lattices: Emergence of Novel Features in Extended States
The system of a cold atomic gas in an optical lattice is governed by two
factors: nonlinearity originating from the interparticle interaction, and the
periodicity of the system set by the lattice. The high level of controllability
associated with such an arrangement allows for the study of the competition and
interplay between these two, and gives rise to a whole range of interesting and
rich nonlinear effects. This review covers the basic idea and overview of such
nonlinear phenomena, especially those corresponding to extended states. This
includes "swallowtail" loop structures of the energy band, Bloch states with
multiple periodicity, and those in "nonlinear lattices", i.e., systems with the
nonlinear interaction term itself being a periodic function in space.Comment: 39 pages, 21 figures; review article to be published in a Special
Issue of Entropy on "Non-Linear Lattice
Approximation of small-amplitude weakly coupled oscillators with discrete nonlinear Schrodinger equations
Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are
approximated by equations of the discrete nonlinear Schrodinger type. We show
how to justify this approximation by two methods, which have been very popular
in the recent literature. The first method relies on a priori energy estimates
and multi-scale decompositions. The second method is based on a resonant normal
form theorem. We show that although the two methods are different in the
implementation, they produce equivalent results as the end product. We also
discuss applications of the discrete nonlinear Schrodinger equation in the
context of existence and stability of breathers of the Klein--Gordon lattice
Transport of gaussian measures by the flow of the nonlinear Schr\"odinger equation
We prove a new smoothing type property for solutions of the 1d quintic
Schr\"odinger equation. As a consequence, we prove that a family of natural
gaussian measures are quasi-invariant under the flow of this equation. In the
defocusing case, we prove global in time quasi-invariance while in the focusing
case because of a blow-up obstruction we only get local in time
quasi-invariance. Our results extend as well to generic odd power
nonlinearities.Comment: Presentation improve
Solitons in nonlinear lattices
This article offers a comprehensive survey of results obtained for solitons
and complex nonlinear wave patterns supported by purely nonlinear lattices
(NLs), which represent a spatially periodic modulation of the local strength
and sign of the nonlinearity, and their combinations with linear lattices. A
majority of the results obtained, thus far, in this field and reviewed in this
article are theoretical. Nevertheless, relevant experimental settings are
surveyed too, with emphasis on perspectives for implementation of the
theoretical predictions in the experiment. Physical systems discussed in the
review belong to the realms of nonlinear optics (including artificial optical
media, such as photonic crystals, and plasmonics) and Bose-Einstein
condensation (BEC). The solitons are considered in one, two, and three
dimensions (1D, 2D, and 3D). Basic properties of the solitons presented in the
review are their existence, stability, and mobility. Although the field is
still far from completion, general conclusions can be drawn. In particular, a
novel fundamental property of 1D solitons, which does not occur in the absence
of NLs, is a finite threshold value of the soliton norm, necessary for their
existence. In multidimensional settings, the stability of solitons supported by
the spatial modulation of the nonlinearity is a truly challenging problem, for
the theoretical and experimental studies alike. In both the 1D and 2D cases,
the mechanism which creates solitons in NLs is principally different from its
counterpart in linear lattices, as the solitons are created directly, rather
than bifurcating from Bloch modes of linear lattices.Comment: 169 pages, 35 figures, a comprehensive survey of results on solitons
in purely nonlinear and mixed lattices, to appear in Reviews of Modern
Physic
A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation
We present the recent result in [29] concerning strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap solutions of the defocusing cubic nonlinear Schrodinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the H-s topology (0 < s < 1) and whose H-s norm can grow by any given factor
Quasi-periodic solutions for 1D Schrödinger equation with the nonlinearity |u|2pu
AbstractIn this paper, one-dimensional (1D) nonlinear Schrödinger equationiut−uxx+|u|2pu=0,p∈N, with periodic boundary conditions is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions corresponding to 2-dimensional invariant tori of an associated infinite-dimensional dynamical system. The proof is based on infinite-dimensional KAM theory, partial normal form and scaling skills
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