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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. - PowerPoint PPT Presentation
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Nonlinear Localised Excitations in the Gap spectrum
Bishwajyoti DeyDepartment of Physics,
University of Pune, Pune
With Galal AlakhalyGA, 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 Dey et al, PRE-2000; Gorbach and Flach, PRE 2005, Kevrikidis, konotop, PRE 2002
Compact-like discrete breather (Eleftheriou, Dey, Tsironis, PRE, 2000)
V(u) is nonlinear onsite potential.
Double well
Hard phi-4 potential
Morse potential
stable 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.
Xu et alElena et al Phys. Rev. Lett 90, 160407 (2003)
BEC inopticallattice
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
Linearize equation of motion around classical ground state
For nonlinear lattice, onsite potential can be nonlinear, or W (intersite interaction) can be nonlinear (anharmonic) or both can be nonlinear.
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 BECYakimenko et alarXiv:1112.6006Dec 2011
GA, BD PRE (2011)
The uncoupled equations ( ) has compacton solutions
Where 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 gapcompacton solutions if they exist in the gapregion will be stableagainst the decay by radiation by resonating with the linear oscillatorywaves.
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 functionsto 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