existence and stability of discrete breathers in a hexagonal lattice application in a dusty plasma...
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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.