Localized oscilla ions in nonlinea hamil onian Klein-Go don la ices. B ea he s and Ande son
modes
J. Cue as, F. Palme o, J.F.R. A chilla, F.R. Rome o. Nonlinea Physics G oup. Uni e si y o Se illa
M.C. Mu iel. Bi u ca ion Theo y and Dynamical Sys ems G oup.Uni e si y o Cádiz
Pi ch o k+Pe iod doubling. q=1 pa h (2d)
In oduc ion
• The e a e wo di e en sou ces o localiza ion in
disc e e la ices:
• Ande son modes in diso de ed ha monic
la ices [1]
• Disc e e b ea he s in homogeneous nonlinea
la ices [2]
Objec i e
• S udy o he condi ions o which localized modes
exis s in diso de ed anha monic la ices
• We unde ake he p oblem es udying he possibili y o
connec ion o disc e e b ea he wi h Ande son modes.
Model
nnn
nnnnn
N
Nn
suuuV
uuCuVumH
−=
−++= +
−=
∑
2
n
2
1
2
ω
2
1
)(
)(
2
1
)(
2
1&
unc ion) (pa h 0,1)(
ρ
ec o ) andom:(
2
)(
ρ
1
ω
>−=
+=
qss
s
q
n
n
n
Connec ion o disc e e b ea he s and
Ande son modes
A solu ion in one o he limi s is calcula ed and
con inued o he o he limi keeping he ac ion
(phase space a ea) cons an .
• The numbe o disc e e b ea he s is huge
compa ed o he numbe o Ande son modes.
• This ac sugges ha he bi u ca ions in he
pa h om b ea he s o Ande son modes
should be u ning poin s and pi ch o ks.
• I also appea s pe iod doubling bi u ca ions
• The Ande son modes o highes and lowes
equency a e connec ed
• I has also been ound he exis ence o isolas
in he las case
• The andom ec o akes i s alues in a
disc e e andom dis ibu ion
Re e ences
1. PW Ande son. Phys Re 109 (1958) 1942
2. S Flach and CR Willis. Phys Rep 295 (1998) 181
3. FR A chilla, RS MacKay and JL Ma ín. Phys D 134
(1999) 406
4. J Cue as, JFR A chilla, F Palme o and FR Rome o.
Jou Phys A 34 (2001) L1
s=0: Linea diso de ed limi (Ande son
modes)
s=1: Nonlinea o de ed limi (disc e e
b ea he s)
B oken pi ch o k in he q=1/4 pa h (2d)
Tu ning poin . q=1 pa h (2d)
Isola. q=1/4 pa h (1d)