Nonlinear Localised Excitations in the Gap spectrum

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

2. 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.

3. 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

4. Compact-like discrete breather(Eleftheriou, Dey, Tsironis, PRE, 2000) V(u) is nonlinear onsite potential. stable unstable Double well Hard phi-4 potential Morse potential

5. 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

6. Xu et al

7. Origin of gap in the spectrum 2. Discrete lattice: Example: BEC amplitude equation for the condensate on a deep optical lattice.

8. 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

9. 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

10. GA, BD PRE (2011)

12. The uncoupled equations ( ) has compacton solutions Where for

13. 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

16. 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.

17. 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

18. 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

19. In terms of new variables, The equations for first and zeroth harmonics can be written as

20. 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

21. Integrating we get Which gives And the equation for R as where the phases satisfy the coupled equations

22. The equation can be written in the compacton equation of the form where

23. Gap soliton solutions Gap compacton-like solutions

24. Finally the solutions can be written in terms of the original field u(x,t) and v(x,t) as