8 research outputs found
Surface Gap Soliton Ground States for the Nonlinear Schr\"{o}dinger Equation
We consider the nonlinear Schr\"{o}dinger equation , with and and with periodic in each coordinate direction. This problem
describes the interface of two periodic media, e.g. photonic crystals. We study
the existence of ground state solutions (surface gap soliton ground
states) for . Using a concentration compactness
argument, we provide an abstract criterion for the existence based on ground
state energies of each periodic problem (with and ) as well as a more practical
criterion based on ground states themselves. Examples of interfaces satisfying
these criteria are provided. In 1D it is shown that, surprisingly, the criteria
can be reduced to conditions on the linear Bloch waves of the operators
and .Comment: definition of ground and bound states added, assumption (H2) weakened
(sign changing nonlinearity is now allowed); 33 pages, 4 figure
Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross-Pitaevskii Equation
We rigorously analyze the bifurcation of stationary so called nonlinear Bloch
waves (NLBs) from the spectrum in the Gross-Pitaevskii (GP) equation with a
periodic potential, in arbitrary space dimensions. These are solutions which
can be expressed as finite sums of quasi-periodic functions, and which in a
formal asymptotic expansion are obtained from solutions of the so called
algebraic coupled mode equations. Here we justify this expansion by proving the
existence of NLBs and estimating the error of the formal asymptotics. The
analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In
addition, we illustrate some relations of NLBs to other classes of solutions of
the GP equation, in particular to so called out--of--gap solitons and truncated
NLBs, and present some numerical experiments concerning the stability of these
solutions.Comment: 32 pages, 12 figures, changes: discussion of assumptions reorganized,
a new section on stability of the studied solutions, 15 new references adde
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
The Discontinuous Galerkin Method for Maxwell\u27s Equations: Application to Bodies of Revolution and Kerr-Nonlinearities
Die unstetige Galerkinmethode (UGM) wird auf die rotationssymmetrischen und Kerr- Maxwell-Gleichungen angewandt. Essentiell ist hierbei der numerische Fluss. Für die rotationssymmetrischen Maxwell-Gleichungen wird ein exakter Fluss vorgestellt und unter Ausnutzung der Symmetrie der Aufwand reduziert. Für die Kerr-Maxwell-Gleichungen führt der exakte numerische Fluss auf eine ineffiziente UGM, weswegen approximative Flüsse miteinander verglichen werden. Wir erhalten optimale Konvergenz