Transcript Slide 1

Nonlinear Localised Excitations
in the Gap spectrum
Bishwajyoti Dey
Department of Physics,
University of Pune, Pune
With Galal Alakhaly
GA, BD Phys. Rev. E 84, 036607 (1-9) 2011
Nonlinear localised excitations – solitons, breathers, compactons.
These solutions are nonspreading – retain their shape in time.
Solitons, breathers and compactons form if the nonlinear dynamics is balanced by the
spreading due to linear dispersion.
For discrete systems the localization is due to the discreteness combined with the
nonlinearity of the system.
For linear systems, the discrete translational invariance have to be broken (adding impurity)
to obtain spatially localized mode (Anderson Localization).
For nonlinear systems one can retain discrete translational symmetry and still obtain
localized excitations. Self localised solutions.
Bright solitons have been observed in BEC where the linear spreading due to dispersion is
compensated by the attractive nonlinear interactions between the atoms.
Compactons – Soliton with compact support
Rosenau and Hyman, PRL, 1993
Dey, PRE, 1998
Solutions stable –
Linear stability, nonlinear
Stability (Lyapunov).
Dey, Khare PRE 1999
Compact-like discrete breathers
Kevrikidis, konotop, PRE 2002
Dey et al, PRE-2000;
Gorbach and Flach, PRE 2005,
Compact-like discrete breather (Eleftheriou, Dey, Tsironis, PRE, 2000)
V(u) is nonlinear onsite potential.
stable
Double well
Morse potential
unstable
Origin of the gap in the spectrum:
1. Presence of periodic potential .
Example: BEC in a periodic potential. Presence of periodic potential leads to the
modification of the linear propagation, dispersion relation. Spectrum of atomic Bloch
waves in the optical lattice is analogous to single electron states in crystalline solids.
Elena et al Phys. Rev. Lett 90, 160407 (2003)
Xu et al
BEC in
optical
lattice
Xu et al
Origin of gap in the spectrum
2. Discrete lattice:
Example: BEC amplitude equation for the condensate on a deep optical lattice.
The Lattice Problem : nonlinear lattice
• Spatial discreteness and Nonlinearity
For nonlinear lattice, onsite potential can be nonlinear, or W (intersite
interaction) can be nonlinear (anharmonic) or both can be nonlinear.
Linearize equation of motion around classical ground state
Origin of gap in the spectrum
3. Coupled nonlinear dynamical evolution equation
Example: (i) Spinor condensates
(ii) Multi species BEC
Soliton in
Binary mixture of BEC
Yakimenko et al
arXiv:1112.6006
Dec 2011
GA, BD PRE (2011)
The uncoupled equations (
Where
) has compacton solutions
for
Existence of the gap
To show that in the systems linear spectrum opened by weak coupling and to find
the width of the gap
Consider the uncoupled linear equations as
The gap soliton or gap
compacton solutions
if they exist in the gap
region will be stable
against the decay by
radiation by resonating
with the linear oscillatory
waves.
Dynamics of the system inside the spectral gap region
To look for localised solutions inside the gap spectrum we consider weak nonlinearity
and assume that the amplitude of U and V are small and slowly varying.
We also assume that the differentiation of slowly varying functions
to be order of coupling constant
Substituting in the coupled equations we get the amplitudes of the second harmonics as
The equations amplitudes of the first harmonics as
and the equations amplitudes of the zeroth harmonics as
In terms of new variables,
The equations for first and zeroth harmonics can be written as
Look for travelling solitary wave solutions – transform to travelling coordinate
We get system of coupled differential equations for for the first harmonics
amplitudes A and B as
The zeroth harmonic amplitudes are given by
Integrating we get
Which gives
And the equation for R as
where the phases satisfy the coupled equations
The equation can be written in the compacton equation of the form
where
Gap soliton solutions
Gap compacton-like solutions
Finally the solutions can be written in terms of the original field u(x,t) and v(x,t) as