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Mathematisches
Kolloquium
Institut für Mathematik
Vortrag im Rahmen des
mathematischen Kolloquiums:
Herr Dr. Tomas Donal
(Universität Karlsruhe)
„Surface Gap Solitions at a Nonlinearity
Interface in the Periodic Schroedinger
Equation“
Abstract:
siehe Rückseite
Ort: Universität Oldenburg
Campus Wechloy
(Carl-von-Ossietzky-Straße)
Raum W1 0-006
Zeit: Mittwoch, den 25. Juni 2008,
17 Uhr c.t.
Kaffee/Tee 16.45 Uhr im Raum
W1 2-213
Zu dieser Veranstaltung laden wir Sie
herzlich ein.
Surface Gap Solitons at a Nonlinearity Interface in the
Periodic Schrdinger Equation
Tom´aˇs Dohnal
Department of Mathematics, University of Karlsruhe, Germany
email: dohnal@math.uka.de
Abstract: In the first part of the talk I will present a study of solitary waves localized
at the interface of two nonlinear periodic media with different coefficients of the cubic
nonlinearity in the one and two-dimensional periodic Schrdinger equation
iut + △u − V (x)u + Γ(x)|u|2
u = 0, x ∈ Rd
, d = 1, 2
Γ = Γ− for x1 ≤ 0, Γ = Γ+ for x1 > 0,
where V is 2π periodic in each variable and Γ+ = Γ− are real constants. The model
is applicable in the field of nonlinear photonic crystals as well as in Bose-Einstein
Condensates (BECs). In photonics applications such a structure corresponds to a
cubically nonlinear photonic crystal with different values of the nonlinear refractive
index on each side of the interface x2 = 0 and in BECs it describes a condensate with
different s−wave scattering lengths on each side of the interface.
We call solutions u(x, t) = e−iωt
φ(x) that are exponentially localized in space surface
gap solitons (SGSs) [1,2] as they are inherent to the interface surface and because their
propagation constant (or frequency) ω lies in the band gaps of the corresponding linear
operator L = −△ + V (x). In our construction SGSs are computed via bifurcation
from standard gap solitons (GSs) with Γ ≡ const. The SGS continuation curves
undergo folds, the localtion of wich is studied analytically in a certain asymptotic
regime. Interesting (unexpected) phenomena such as concentration of the SGSs in
the less focusing half of the medium are observed.
In the second half of the talk we study the one dimensional system with an interface
that is linear as well as nonlinear. The two scenarios of a linear interface we consider
are a jump in the value and derivative of V (x) at x1 = 0. We show under which
condition each case leads to the occurence of point spectrum of L. SGSs then become
nonlinear defect modes.
Acknowledments: This work is in partial collaboration with Dmitry Pelinovsky (McMaster University, Canada), Michael Plum and Wolfgang Reichel (both: University
of Karlsruhe, Germany).
References:
1. Y.V. Kartashov, A.A. Egorov, V.A. Vysloukh and L. Torner, Opt. Express 14,
4049–4057 (2006).
2. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R.
Morandotti, H. Yang, G. Salamo and M. Sorel, Phys. Rev. Lett. 96, 063901
(2006).
3. T. Dohnal and D. Pelinovsky, ”Surface gap solitons at a nonlinearity interface,”
SIAM J. Appl. Dyn. Syst. 7, 249–264 (2008).
Oldenburg-Invitation to the Math-Coloquium at 6/25/08 Oldenburger Einladung zum Mathe-Kolloquium am 25.6.08

 



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