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