Existence and Stability of Discrete Breathers

in a Hexagonal Lattice

Application in a Dusty Plasma Crystal

V. Koukouloyannis, I. Kourakis

Hamiltonian Lattice Dynamical Systems Leiden, October 2007

Hamiltonian Lattice Dynamical Systems Leiden, October 2007

Existence of 3-site breathers in a Hexagonal Lattice

22 2

0 1 1, 1, 1, ,

2 2 2 2, 1 , 1 1, 1 1,

( ) [( ) ( )2 2

( ) ( ) ( ) ( ) ]

ijij ij i j ij i j

i j i j

ij i j ij i j ij i j ij i j

pH H H V x x x x x

x x x x x x x x

1 0i

H

1 1

1.H H dt

T

0

( ) ( )cos( ),nn

x t A J nw

1 12 21 2 2 3

1 11 2

1 1(sin sin ) 0, (sin sin ) 0

2 2n nn n

H HnA n n nA n n

1 2 13 3 1 2 1

2 3 1

w ww w

w w

1 2 1 2 1 2

2 40, 0 ,

3 3

This system is described by the Hamiltonian

Consider the phase differences

(VK and R.S.MacKay, J. Phys. A, 38 (2005) 1021-1030)This system supports 3-site breathers if

where Since

which always has the solutions

Hamiltonian Lattice Dynamical Systems Leiden, October 2007

The single-site breather

Up to tree moving sites there are four cases of breathers

Case (a): Single site breather

Hamiltonian Lattice Dynamical Systems Leiden, October 2007

The 3-site breathers

Case (b): 1 2 0 In-phase 3-site breather

0J

Linear stability condition for small ε:

Hamiltonian Lattice Dynamical Systems Leiden, October 2007

The 3-site breathers

Case (c): 1 20 Out of phase 3-site breather

Linear stability condition for small ε: 0J

2 2

1

( 1) 0nn

n

n A

and

Hamiltonian Lattice Dynamical Systems Leiden, October 2007

The 3-site breathers

Condition for linear stability for small ε:

1 2

2 4

3 3

Case (d): Vortex 3-site breather

0fJ

with

2 2

1

( 1) cos3

nn

n

nf n A

Hamiltonian Lattice Dynamical Systems Leiden, October 2007

The dusty plasma crystal

Table 1: Values suggested by A. Melzer

a b ε

set I 0.01 -0.04 0.034

set II 0.01 -0.06 0.065

set III -0.21 -0.02 0.17

The transverse displacement is described by the Klein-Gordon Hamiltonian

0J

with ε<0 (inverse dispersion)

and 2 3 4

( )2 3 4

x x xV x a b

Since

and the vortex breather are unstable even for small ε.

, the out of phase

For set I the single and the in phase breathers can be continued for large enough ε so the system can support them.