Using he ini e domain emnan o he con inuous spec um o examine in eg abili y:
E ec o bounda y condi ions
Panayo is G. Ke ekidis1and Niu ka R. Quin e o2
1Depa men o Ma hema ics and S a is ics, Uni e si y o Massachuse s, Amhe s , Massachuse s 01003-4515, USA
2Depa amen o de Fı
´sica Aplicada I, Escuela Uni e si a ia Poli e
´cnica, Uni e sidad de Se illa, Vi gen de A
´ ica 7, 41011 Se illa, Spain
and Ins i u o Ca los I de Fı
´sica Teo
´ ica y Compu acional, Uni e sidad de G anada, E-18071 G anada, Spain
共Recei ed 11 Decembe 2002; e ised manusc ip ecei ed 30 June 2003; published 23 Sep embe 2003兲
The aim o his wo k is o p opose a me hod o es ing he in eg abili y o a model pa ial di e en ial 共PDE兲
and/o di e en ial di e ence equa ion 共DDE兲, by examining i in a ini e bu la ge domain. Fo monopa ame -
ic amilies o PDE/DDE’s, ha a e known o possess isola ed in eg able poin s, we ind ha e y special
ea u es occu in he ini e domain emnan o he con inuous 共‘‘phonon’’兲spec um a hese ‘‘singula ’’ poin s.
We iden i y hese ea u es in he case example o a PDE and a DDE 共 ha sus ain on and pulselike solu ions,
espec i ely兲 o di e en ypes o bounda y condi ions. The key inding o he wo k is ha such spec al
ea u es a e gene ic nea he singula , in eg able poin s and hence we p opose o explo e a gi en PDE/DDE in
a ini e bu la ge domain o such ai s, as a means o assessing i s po en ial in eg abili y.
DOI: 10.1103/PhysRe E.68.036612 PACS numbe 共s兲: 05.45.⫺a, 02.30.Ik
I. INTRODUCTION
In eg able models o pa ial di e en ial 共PDE兲and di e -
en ial di e ence 共DDE兲equa ions ha e been a opic o in-
ense in es iga ion o e he pas ew decades 关1–3兴. The
main eason o his, excep o he wide a ie y o physical
applica ions ha can be desc ibed by in eg able o nea -
in eg able sys ems, is ha he special case o in eg able mod-
els can be analyzed comple ely by means o he in e se sca -
e ing ans o m 关1,4兴. This can hen se e as a s a ing poin
o pe u ba i e ea men o nea -in eg able sys ems.
In he p ocess o hese de elopmen s, a numbe o ech-
niques ha e been de eloped o assessing in eg abili y in
con inuous 关5兴o disc e e 关6兴se ings 共o applicable o bo h
关7兴兲. An in e es ing ea u e o hese ‘‘ es s’’ is ha hey a e
necessa y 共bu no su icien 兲condi ions o in eg abili y.
Hence, i a model equa ion ails such a c i e ion, i is nonin-
eg able, bu i i passes, i may o may no be in eg able. In
a sense, his sugges s ha we s ill do no unde s and he
essen ial ing edien s ha ende a sys em comple ely in e-
g able. O cou se, should a Lax pai be iden i ied and he
in e se sca e ing mechanism be applied, we know ha he
sys em is in eg able, bu i would ce ainly be desi able 共as is
clea om all he abo e e o o c ea e ‘‘in eg abili y es s’’兲
o ha e a mechanis ic 共‘‘black box’’兲 ype o c i e ion o
assess ha .
We, o cou se, do no claim o be p o iding a ull answe
o his ques ion in he p esen wo k. Howe e , we will y o
gi e a numbe o use ul hin s ha may lead o pa ial an-
swe s o he abo e ques ions and may p o ide some in ui ion
in he e o o cons uc such mechanis ic c i e ia.
Ou ool o choice will be he use o di e en se s o
bounda y condi ions 共BC兲 o examine he spec um o he
linea iza ion a ound he nonlinea cohe en s uc u e ha he
PDE/DDE o in e es suppo s. No ice ha he e ec o
bounda y condi ions in ela ed con ex s has been s udied in a
numbe o e e ences; see, e.g., Re . 关8兴, and e e ences
he ein. Howe e , in all o hese wo ks he e ec s o he BC
o he poin spec um we e assessed and mo eo e , his was
no done in di ec connec ion wi h issues o in eg abili y.
He e we will, ins ead, ocus on he con inuous spec um; in
ac , since we will be dealing wi h ini e bu la ge domains,
we will cen e ou a en ion a ound he disc e e spec um
emnan ha ‘‘becomes’’ he con inuous spec um in he in-
ini e domain limi . In he ini e domain case, he 共 o me ly
con inuous兲spec um becomes disc e e due o he quan iza-
ion o he wa e numbe s, imposed by he bounda y condi-
ions 共see, e.g., Sec. II below兲. I is exac ly his disc e e em-
nan o he con inuous spec um, ha we aim a examining
he e, o elucida e i s in e es ing p ope ies in in eg able e -
sus nonin eg able se ings.
In he p esen wo k, we ocus on wo model p oblems, o
es ablish ou indings and demons a e hei gene ali y. The
models a e selec ed as one-pa ame e amilies o equa ions
such ha one membe o he amily is an in eg able sys em.
Mo eo e , in illus a ing he gene ali y o he conclusions,
hey a e selec ed in a o m such ha one model co esponds
o a PDE, while he o he o a DDE, so ha one is kink
bea ing, while he o he is pulse bea ing. The models o in-
e es will be he pa ame ically modi ied sine-Go don equa-
ion 关o en also called he Pey a d-Remoissene 共PR兲model兴
关9,10兴and a modi ied e sion o he disc e e nonlinea
Sch o
¨dinge 共DNLS兲model 共occasionally called he Sale no
model兲关11兴. The o me PDE eads
␾
⫺
␾
xx⫽⫺ dU
d
␾
,U共
␾
, 兲⫽共1⫺ 兲2关1⫺cos共
␾
兲兴
1⫹ 2⫹2 cos共
␾
兲
共1兲
in he in ini e domain
兩
x
兩
⬍⬁and wi h
兩
兩
⬍1; while he
la e DDE is o he o m
iu
˙n⫽⫺⌬2un⫺
兩
un
兩
2关2
⑀
un⫹共1⫺
⑀
兲共un⫹1⫹un⫺1兲兴.共2兲
The mos well known among hese monopa ame ic amilies
o models a e he sine-Go don equa ion 关Eq. 共1兲, o ⫽0]
which is ele an o supe conduc i i y and cha ge densi y
PHYSICAL REVIEW E 68, 036612 共2003兲
1063-651X/2003/68共3兲/036612共7兲/$20.00 ©2003 The Ame ican Physical Socie y68 036612-1
wa es among o he applica ions 关2兴and he expe imen ally
ealizable disc e e nonlinea Sch o
¨dinge equa ion 关12兴o
⑀
⫽1, as well as i s in eg able, so-called Ablowi z-Ladik 关13兴
coun e pa o
⑀
⫽0 in he case o Eq. 共2兲.
No ice ha o he PDE, he subsc ip s deno e pa ial de-
i a i es o he ield, while o he DDE, he o e do deno es
empo al de i a i e, ⌬2un⬅C(un⫹1⫺2un⫹un⫺1), whe e
C⫽1/(⌬x)2is a cons an de e mined by he la ice spacing
⌬x; he subsc ip ndeno es he la ice si e index. In he
o me case, he e exis kinklike solu ions which ha e been
de ailed in Re s. 关9,10兴, while in he la e , he ield is com-
plex and he e exis pulselike solu ions o he o m un
⫽exp(i⌳ ) n, whe e ⌳is he equency o he solu ions and
ni s 共 eal兲exponen ially localized spa ial p o ile 关11,12兴.
In he PDE, linea iza ion a ound a s a e
␾
0(x), using he
ansa z
␾
⫽
␾
0(x)⫹
␦
exp(i
␻
) (x) in o Eq. 共1兲, yields o
O(
␦
) he linea iza ion equa ion
xx⫹关
␻
2⫺U⬙共
␾
0, 兲兴 ⫽0. 共3兲
No ice ha when ⫽0共in he in ini e domain limi 兲,
␾
0(x)
⫽4 a c an关exp(x)兴is he s a ic kink solu ion o he sG equa-
ion and o his unc ion, he S u m-Liou ille p oblem 共3兲
can be exac ly sol ed 关14兴yielding one disc e e mode 共Gold-
s one mode兲a
␻
⫽0 and he con inuous spec um ep e-
sen ed by he phonons,
␻
k⫽
冑
1⫹k2, k共x兲⫽exp共ikx兲
冑
2
␲␻
k
关k⫹i anh共x兲兴,共4兲
o all alues o k. Fo ⫽0, nei he he s a ic solu ion no
he linea iza ion spec um a e explici ly a ailable in he in i-
ni e domain limi .
Analogously o he PDE, o he linea s abili y analysis
o DDE 共2兲we inse exp(i⌳ )关 n⫹
␦
(Une⫺i
␻
⫹Wnei
␻
쐓 )兴in o
Eq. 共2兲. We hus ob ain o O(
␦
) he ollowing eigen alue
p oblem o 兵
␻
,
兵
Un,Wn
쐓
其
其:
␻
冉
Un
Wn
쐓
冊
⫽L
冉
Un
Wn
쐓
冊
,L⫽
冉
AB
⫺B⫺A
冊
,
Amn⫽关⌳⫹2C⫺
兵
4
⑀
n
2⫹共1⫺
⑀
兲 n关 n⫹1⫹ n⫺1兴
其
兴
␦
m,n
⫹关共1⫺
⑀
兲 n
2⫺C兴共
␦
m,n⫹1⫹
␦
m,n⫺1兲,
Bmn⫽⫺ n关2
⑀
n⫹共1⫺
⑀
兲共 n⫹1⫹ n⫺1兲兴
␦
m,n,共5兲
whe e he s a s deno e complex conjuga ion.
The pape is o ganized as ollow. In he ollowing sec ion
we ob ain an app oxima e solu ion o he S u m-Liou ille
p oblem 共3兲by imposing di e en ypes o bounda y condi-
ions in he ini e domain o leng h L. The ob ained esul s
a e compa ed wi h he nume ical compu a ions in Sec. III,
whe e we also compu ed he solu ion o Eq. 共5兲. Finally, we
summa ize ou indings and p esen ou conclusions in Sec.
IV.
II. ANALYTICAL APPROXIMATION
In his sec ion we sol e app oxima ely Eq. 共3兲when
兩
x
兩
⬍L/2, whe e Lis he ini e 共bu la ge enough兲leng h o he
sys em. No ice ha ou esul s will be gene ically ue, i Lis
chosen la ge enough. By la ge enough he e, we mean a do-
main size which is many imes 共a leas 10兲la ge han he
cha ac e is ic leng h o he soli a y wa e 共kink o pulse兲 ha
we will examine inside his domain. We will ake in o ac-
coun di e en kinds o BC, in pa icula , ee
x共⫺L/2兲⫽0, x共L/2兲⫽0, 共6兲
ixed
共⫺L/2兲⫽0, 共L/2兲⫽0, 共7兲
and an ipe iodic bounda y condi ion 共aPBC兲
x共⫺L/2兲⫽⫺ x共L/2兲, 共⫺L/2兲⫽⫺ 共L/2兲.共8兲
Fi s we conside he in eg able case, ⫽0, and we show ha
o he i s phonon modes, he eigen equencies
␻
˜
n
ee
⫽
␻
˜
n⫺1
ixed and
␻
˜
n
ap ha e a double mul iplici y 共we will deno e
wi h ilde he analy ical, app oxima ed eigen equencies兲.To
p oceed, we use he exac solu ion o p oblem 共3兲 o ⫽0in
he in ini e domain. We would like o s ess ha i we change
he in ini e domain by a ini e one, wi h a gi en BC, we will
s ill ha e an in ini e numbe o eigen equencies,1bu o he
allowed wa e numbe s k关15兴. In o de o calcula e app oxi-
ma ely hese allowed wa e numbe s, we p oceed as in Re .
关16兴. No ice ha k(x)⫽Fk(x)⫹iGk(x), whe e
Fk共x兲⫽kcos共kx兲⫺sin共kx兲 anh共x兲
冑
2
␲␻
k
,共9兲
Gk共x兲⫽cos共kx兲 anh共x兲⫹ksin共kx兲
冑
2
␲␻
k
.共10兲
Then he solu ion o Eq. 共3兲, wi h ⫽0, ela ed o he pho-
non con ibu ion is ep esen ed by he linea supe posi ion o
all he odd 关Gk(x)兴and e en 关Fk(x)兴phonon modes
␺
共x, 兲⫽兺
k关ak共 兲Fk共x兲⫹bk共 兲Gk共x兲兴.共11兲
Imposing ee BC o each phonon mode o Eq. 共11兲we
ob ain ha he i s wa e numbe s sa is y
ak共 兲关sin共kL/2兲关k2⫹cosh⫺2共L/2兲兴⫹kcos共kL/2兲 anh共L/2兲兴
⫽0, 共12兲
bk共 兲关cos共kL/2兲关k2⫹cosh⫺2共L/2兲兴⫺ksin共kL/2兲 anh共L/2兲兴
⫽0. 共13兲
1No e ha his is ue o he con inuum p oblem o Eq. 共3兲, bu
would no longe be ue o he disc e e one o Eq. 共5兲.
P. G. KEVREKIDIS AND N. R. QUINTERO PHYSICAL REVIEW E 68, 036612 共2003兲
036612-2
The solu ions o hese anscenden al equa ions yield he al-
lowed alues o k. We can sol e hese app oxima ely i we
conside LⰇ1. Then, we ind ha
kn,0
ee⫽n⫺1
L
␲
,n⫽1,2,3,..., nⰆL,共14兲
whe e he ze o subsc ip deno es ha we a e dealing wi h he
unpe u bed case ⫽0, and i s co esponding eigen unc ions
a e ela ed wi h he odd unc ions Gn(x) o he odd numbe s
nand wi h he e en unc ions Fn(x) o he e en numbe s n.
Hence, he i s eigen equencies a e ep esen ed by
␻
˜
n,0
ee⫽
冑
1⫹
冉
n⫺1
L
␲
冊
2
,n⫽1,2,..., nⰆL.
共15兲
Analogously, o ixed BC he ollowing ela ions hold;
ak共 兲关kcos共kL/2兲⫺sin共kL/2兲 anh共L/2兲兴⫽0, 共16兲
bk共 兲关cos共kL/2兲 anh共L/2兲⫹ksin共kL/2兲兴⫽0. 共17兲
Then, o la ge enough L, we ind ha
kn,0
ixed⫽n
L
␲
,n⫽1,2,3,..., nⰆL,共18兲
and so,
␻
˜
n,0
ixed⫽
冑
1⫹
冉
n
L
␲
冊
2
,n⫽1,2,..., nⰆL,共19兲
whe e he odd 共e en兲numbe s na e ela ed wi h he odd
Gn(x)关e en Fn(x)] eigen unc ions.
Rema k 1. By compa ing exp essions 共15兲and 共19兲we
obse e ha in he in eg able case ( ⫽0)
␻
˜
n
ee⫽
␻
˜
n⫺1
ixed o
he i s ew eigen equencies.
Now by imposing aPBC in each phonon mode o Eq. 共11兲
and aking in o accoun he symme y p ope ies o Fk(x),
Gk(x) and hei de i a i es, he equa ions ha he wa e
numbe sa is ies can be educed o
akFk共L/2兲⫽0, 共20兲
bk
⳵
Gk
⳵
x共L/2兲⫽0. 共21兲
No ice ha Eqs. 共20兲and 共21兲coincide wi h Eqs. 共16兲共i.e.,
he i s equa ion o ixed BC兲and 共13兲共i.e., he second
equa ion o ee BC兲, espec i ely. The solu ions o Eqs.
共20兲and 共21兲a e gi en by
kn,0
ap⫽2共n⫺1兲
␲
L,n⫽2,3,..., nⰆL,共22兲
kn,0
ap⫽2共n⫺1兲
␲
L,n⫽1,2,3,..., nⰆL,共23兲
espec i ely, and hei eigen unc ions co espond o he e en
Fn(x) and odd Gn(x). Then, he i s eigen equencies a e
ep esen ed by
␻
˜
n,0
ap⫽
冑
1⫹
冉
2共n⫺1兲
␲
L
冊
2
,n⫽2,3..., nⰆL,共24兲
␻
˜
n,0
ap⫽
冑
1⫹
冉
2共n⫺1兲
␲
L
冊
2
,n⫽1,2,..., nⰆL.
共25兲
This means ha he e en 共odd兲modes o aPBC
兵
␻
˜
n
ap ,an( )Fn(x)
其
关
兵
␻
˜
n
ap ,bn( )Gn(x)
其
兴coincide wi h he
e en modes o ixed BC 共odd modes o ee BC兲.
Rema k 2. F om ela ions 共24兲and 共25兲we conclude ha
o he in eg able case and aPBC he eigen equencies ha e
mul iplici y 2.
The analysis o he S u m-Liou ille p oblem 共3兲 o he
nonin eg able case, ⫽0, becomes mo e complica ed since
␾
0(x) is he exac kink solu ion o Eq. 共1兲and his unc ion
is only known in he implici o m 关9兴共e en o he in ini e
domain p oblem兲. So, ins ead o sol ing his equa ion we
calcula e app oxima ely he solu ion o
冋
d2
dx2⫺V共x兲⫺ W共x兲⫹E
册
⫽0, 共26兲
whe e V(x)⫽⫺2/cosh2(x), W(x)⫽8 anh(x)关x
⫺5 anh(x)兴/cosh2(x), and E⫽
␻
2⫺
␻
ph
2wi h
␻
ph⫽(1
⫺ )/(1⫹ )关10,17兴. This eigen alue p oblem is ob ained in
wo s eps: i s we ind a solu ion o small o Eq. 共1兲,
h ough he pe u ba i e expansion
␾
(x, )⫽
␾
sG(x)
⫹
␾
1(x)⫹O( 2), whe e
␾
sG(x) is he s a ic sG kink and
second we linea ize Eq. 共1兲a ound he ob ained solu ion up
o o de o , so we inse
␾
(x, )⫽
␾
sG(x)⫹
␾
1(x)
⫹
␦
关 (x)exp(i
␻
)⫹ 쐓(x)exp(⫺i
␻
)兴in o Eq. 共1兲and conside
he equa ion ha a ises o O(
␦
) and ob ain Eq. 共26兲. A gu-
ably, his app oach ails o cap u e he co ec ions o he ail
o he wa e due o domain ini eness. Howe e , as a gued in
Re . 关8兴, he la e a e exponen ially small in he leng h o he
domain. Hence, as will also be jus i ied a pos e io i, he e we
cap u e he leading o de dependence in L, as well as he
leading o de e ec o 关see, e.g., Eqs. 共31兲–共33兲below兴.
Then, ollowing he p ocedu e o he pe u ba ion me hods
o linea eigen alue p oblem sugges ed in Re . 关18兴,weas-
sume he solu ion o Eq. 共26兲as
En⫽En,0⫹ En,1⫹O共 2兲,共27兲
n共x兲⫽ n,0共x兲⫹ n,1共x兲⫹O共 2兲,共28兲
whe e he i s subsc ip in he unc ions, n, deno es he o de
o he phonon modes 共 o ⬍0 his subsc ip can also deno e
he in e nal mode兲, he second one co esponds o he o de
o pe u ba ion. By inse ing hese expansions in Eq. 共26兲
and equa ing and collec ing he e ms o he same o de in ,
we ob ain o O( 0),
USING THE FINITE DOMAIN REMNANT OF THE... PHYSICAL REVIEW E 68, 036612 共2003兲
036612-3
冋
d2
dx2⫺V共x兲⫹En,0
册
n,0⫽0, 共29兲
and o he nex o de co ec ion O( 1),
冋
d2
dx2⫺V共x兲⫹En,0
册
n,1⫽关W共x兲⫺En,1兴 n,0 .共30兲
No ice ha Eq. 共29兲co esponds o he in eg able case
⫽0 al eady sol ed o ee 关Eq. 共15兲兴, ixed 关Eq. 共19兲兴, and
an ipe iodic BC 关see Eqs. 共24兲and 共25兲兴. No ice also ha
En,0⫽
␻
˜
n,0
2⫺1⫽kn,0
2and ha i s co esponding eigen unc ion
n,0(x) is ela ed ei he wi h he odd Gn(x) o e en Fn(x).
Then, o di e en bounda y condi ions, he eigen equencies
o Eq. 共26兲a e de e mined by
␻
˜
n
ee⫽
冑
␻
ph
2⫹共kn,0
ee兲2⫹ En,1
ee,共31兲
␻
˜
n
ixed⫽
冑
␻
ph
2⫹共kn,0
ixed兲2⫹ En,1
ixed,共32兲
␻
˜
n
ap⫽
冑
␻
ph
2⫹共kn,0
ap兲2⫹ En,1
ap,共33兲
whe e kn,0
ee ,kn,0
ixed , and kn,0
ap a e gi en by Eqs. 共14兲,共18兲,
and 共22兲and 共23兲, espec i ely.
The solu ion o he eigen alue En,1 o he i s -o de co -
ec ion is gi en by
En,1⫽
冕
⫺L/2
L/2 dx n,0共x兲W共x兲 n,0共x兲
冕
⫺L/2
L/2 dx n,0
2共x兲
.共34兲
The in eg als in ol ed in Eq. 共34兲can be compu ed nume i-
cally o di e en BC and di e en alues o (
兩
兩
Ⰶ1), hen
we can calcula e he app oxima ed eigen equencies in each
case. We can now compa e hese esul s wi h he nume ical
solu ions o Eq. 共3兲关 o de ails on he nume ical me hods/
esul s, we e e he eade o Sec. III兴.
F om he da a o he Tables I and II we obse e an oscil-
la o y beha io o
␻
˜
n
ee⫺
␻
˜
n⫺1
ixed o he i s phonon’s modes
o ⫽0. We also no ice ha he eigen alues o aPBC lose
hei double mul iplici y ha exis ed in he case o he in e-
g able equa ion.
I is also wo h no ing ha hese ea u es a e ypically
obse able in he hi d decimal digi o he co esponding
eigen equencies. On he o he hand, he di e ence 共well jus-
i ied wi hin he app oxima ions men ioned abo e兲be ween
he heo e ical and nume ical p edic ions o he indi idual
eigen equencies is ypically in he ou h o i h decimal
digi . Hence, he obse a ions o he p e ious pa ag aph a e
sys ema ic and in ag eemen wi h he heo e ical p edic ions.
III. NUMERICAL RESULTS AND DISCUSSIONS
To ind he nume ical solu ion o Eqs. 共1兲and 共3兲,we
disc e ize he equa ions in a nume ical mesh o a ini e do-
main. The mesh consis s o he N⫹1 poin s xj⫽
兵
⫺L/2
⫹j⌬x,j⫽0,1,2,...,N
其
de ined in he ini e leng h Lo he
sys em (⌬x⫽L/N). No ice ha since, in his case, we wish
o emula e he beha io o he PDE, ⌬xis e y ine 共 ypi-
cally 0.05), and he obus ness o he indings upon a ia ion
o he 共small兲⌬xhas been e i ied. When we compu e he
solu ion ei he o he PDE o o he linea iza ion equa ion,
we conside h ee di e en ypes o BC 共6兲–共8兲. We would
like o ema k ha his kind o disc e iza ion o he S u m-
Liou ille p oblem 共3兲only a ec s he las phonon modes, so
we can compa e he beha io o he i s phonon modes ob-
ained in he p eceding sec ion wi h he nume ical solu ion o
Eq. 共3兲. In bo h cases, he dis ibu ions o hese eigen e-
quencies a e de e mined by he pa ame e and by he
di e en bounda y condi ions in he ini e domain.
TABLE I. Fo posi i e and small alue o ⫽0.02, we compa e
he i s eigen equencies, ob ained pe u ba i ely,
␻
˜
n, wi h he
ones compu ed by sol ing he o iginal Eqs. 共1兲and 共3兲,
␻
n.
n
␻
n
ee
␻
˜
n
ee
␻
n
ixed
␻
˜
n
ixed
␻
n
ap
␻
˜
n
ap
1 0.96117 0.96192 0.96234 0.96352 0.96117 0.96192
2 0.96132 0.96129 0.96293 0.96284 0.96293 0.96284
3 0.96436 0.96514 0.96729 0.96760 0.96436 0.96514
4 0.96560 0.96541 0.96933 0.96899 0.96933 0.96899
5 0.97118 0.97126 0.97604 0.97573 0.97118 0.97126
6 0.97411 0.97358 0.97989 0.97915 0.97991 0.97915
TABLE II. We p o ide he same compa ison as in he p e ious able o a nega i e alue o ⫽
⫺0.02. He e, ⍀
˜
iand ⍀i ep esen he in e nal mode calcula ed by he pe u ba ion me hod and compu ed by
nume ical solu ion o Eq. 共3兲, espec i ely.
n
␻
n
ee
␻
˜
n
ee
␻
n
ixed
␻
˜
n
ixed
␻
n
ap
␻
˜
n
ap
⍀i⫽1.03560 ⍀
˜
i⫽1.03977 ⍀i⫽1.03560 ⍀
˜
i⫽1.03924 ⍀i⫽1.03560 ⍀
˜
i⫽1.03977
1 1.04131 1.04129 1.04278 1.04271 1.04156 1.04058
2 1.04156 1.04058 1.04367 1.04303 1.04278 1.04271
3 1.04524 1.04506 1.04866 1.04835 1.04694 1.04625
4 1.04694 1.04625 1.05125 1.05057 1.04867 1.04835
5 1.05306 1.05257 1.05838 1.05770 1.05657 1.05564
P. G. KEVREKIDIS AND N. R. QUINTERO PHYSICAL REVIEW E 68, 036612 共2003兲
036612-4
We also compu e he solu ions o DDE 共2兲and Eq. 共5兲
using 200 poin s and ⌬x⫽0.75. The BC a e de ined analo-
gously h ough U0⫽U1,UN⫽UN⫺1,W0⫽W1, and WN
⫽WN⫺1 o ee BC. Fo ixed BC: U0⫽0, UN⫽0, W0
⫽0, and WN⫽0, while o pe iodic BC: U0⫽UN⫺1,UN
⫽U1,W0⫽WN⫺1, and WN⫽W1.
Ou esul s when he pa ame e o he PR po en ial o
⑀
in
he DDE a e a ied can be summa ized in Figs. 1–6.
F om he abo e esul s, he ollowing conclusions can be
d awn.
共1兲Fo ixed BC, he band edge equency is p ohibi ed.
Hence, we compa e
␻
n
ee wi h
␻
n⫺1
ixed . We ind ha o small
wa e numbe s, ixed and ee BC eigen equencies p ac i-
cally coincide only in he in eg able case, whe eas o he
nonin eg able case we obse e an oscilla o y beha io o his
unc ion 关see Figs. 1 and 2兴. In Fig. 2 we also show he
oscilla o y beha io o
␻
˜
n
ee⫺
␻
˜
n⫺1
ixed , ob ained om he pe -
u ba ion heo y, o ⫽⫺0.02 共open iangles兲and ⫽0.02
共open squa es兲.
共2兲Fo an ipe iodic BC, he spec um comp ises o modes
coming al e na ely om he ee and ixed BC. This seems
na u al as he ee bounda y condi ions selec eigenmodes
FIG. 1. Compa ison o he eigen equencies o he 共disc e e
emnan o he兲con inuous spec um o ixed and ee BC: We
ha e plo ed he di e ence be ween he eigen equencies compu ed
om Eq. 共3兲and
␻
ph⫽(1⫺ )/(1⫹ ) s . The ci cles joined by
solid line 共 ee BC兲 ep esen how a he equencies a e om he
lowe phonon mode. The iangles joined by do ed lines co espond
o ixed BC.
FIG. 2. The di e ence be ween he i s equencies o ee and
ixed BC,
␻
n
ee⫺
␻
n⫺1
ixed (2⭐n⭐20), is plo ed as a unc ion o he
wa e numbe o ⫽0共ci cles joined by solid line兲, ⫽⫺0.02 共 i-
angles joined by dashed line兲, and ⫽0.02 共squa es joined by do ed
line兲. The open iangles ( ⫽⫺0.02) and squa es ( ⫽0.02) ep e-
sen he di e ences be ween he equencies ob ained by he pe u -
ba ion heo y,
␻
˜
n
ee⫺
␻
˜
n⫺1
ixed , in he p eceding sec ion.
FIG. 3. An ipe iodic BC: he di e ence be ween he nume ical
eigen equencies compu ed om Eq. 共3兲and he band edge o he
共 o me ly con inuous兲spec um
␻
ph⫽(1⫺ )/(1⫹ ) is shown. Ad-
jacen eigenmodes a e gi en by ci cles joined by solid line and
iangles joined by do ed line. The ele an in e nal mode is shown
by ci cles joined by solid line 共 he i s cu e om below兲.
FIG. 4. An ipe iodic BC: We show he di e ence be ween
␻
n
pe iod⫺
␻
n⫹1
pe iod s n(n⫽2,4,...,20). The s a s p ac ically a
ze o o all n ep esen he in eg able sys em ( ⫽0), whe eas he
long-dashed ( ⫽⫺0.02) and do ed ( ⫽0.02) lines co espond o
nonin eg able cases 共nume ical esul s兲. Wi h iangles ( ⫽
⫺0.02) and ci cles ( ⫽0.02) we plo he eigen equencies ob ained
om he pe u ba ion heo y 共analy ical esul s兲.
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036612-5

symme ic a he bounda y, he ixed ones selec modes an-
isymme ic a he bounda y, while he an ipe iodic BC allow
o bo h 共c . Figs. 3 and 4兲.
共3兲An addi ional ea u e, equally impo an as 共1兲共espe-
cially in iew o i s po en ial p edic i e powe 兲is he ac
ha o he in eg able case o ⫽0, an ipe iodic BC essen-
ially imply he p esence o double eigen alues. The di e -
ence be ween he wo eigen alues is O(10⫺9) o all pai s
共excep o he cu o , disc e iza ion induced phenomena a
he uppe end o he spec um which a e i ele an 兲. This is
in sha p con as 共in pa icula , o small wa e numbe s兲, o
e en mild b eakings o in eg abili y, as can be in e ed om
Fig. 4.
共4兲S a emen s 共1兲and 共3兲abo e can be used in p edic i e
o m and cons i u e he c i e ion 共algo i hm兲se o h in his
wo k: o a gi en PDE/DDE model, we ind he s eady s a e
cohe en s uc u e 共i.e., soli a y wa e兲in a ini e bu la ge
domain. This can be done, e.g., by inding he exac solu ion
o an ODE o nume ically pe o ming a New on- ype algo-
i hm. Linea ize a ound he exac , ini e domain solu ion and
s udy, in pa icula , he small wa e numbe s, close o he
lowe edge o he spec um 共we assume ha he p oblem is
monopa ame ic in wha ollows, bu i is clea ha he ap-
plica ion o he c i e ion does no equi e ha 兲. I o a
c i ical/singula alue o he pa ame e he ixed BC and ee
BC 共small k) eigen alue spec a 共o he emnan o wha o
he in ini e domain was he con inuous spec um兲essen ially
coincide and he mul iplici y o an ipe iodic BC eigen alues
becomes double, hen he model o his unique alue o he
pa ame e can be ‘‘s ongly suspec ed’’ o be in eg able. We
use he abo e exp ession, as we p o ide no igo ous p oo ,
bu only suppo ing 共bu a he uni e sal in dis inc models
wi h dis inc ea u es/solu ions兲nume ical e idence o his
s a emen .
共5兲We ha e also es ed he alidi y o hese esul s in Eq.
共2兲, in he icini y o he in eg able limi
⑀
⫽0, wi h simila
conclusions 关see Figs. 5 and 6兴. Indeed, in Fig. 5 we obse e
he oscilla o y beha io o
␻
n
ee⫺
␻
n⫺1
ixed in he nonin eg able
case, in Fig. 6, we show he case o pe iodic BC, whe e i
can be clea ly seen ha i is only o he in eg able case ha
he double eigen alue mul iplici y is ob ained.
IV. CONCLUSIONS
In conclusion, we ha e p oposed and used a es o e-
ealing he po en ial in eg able na u e o a gi en model p ob-
lem. By a ying he bounda y condi ions o a ini e domain
compu a ion and examining he e ec s o such a ia ions in
he 共con inuous- u ned-disc e e兲spec um, we ha e e ealed
ha he small wa e numbe s ha e singula ways o espond-
ing o he unique pa ame e alues o which he model is
in eg able. These singula ea u es 关such as an app oxima e
iden i ica ion o ixed wi h ee BC o small keigen alues
and he double mul iplici y o eigen alues o pe iodic 共o
an ipe iodic兲BC兴can be used o iden i y and single ou he
in eg able beha io . We ha e p o ided wo model examples,
espec i ely, o kinks and pulses and o a PDE and a DDE.
Independen ly o he de ailed s uc u e o he model hese
p ope ies ha e been iden i ied as uni e sal and ha e been
suppo ed also by analy ical conside a ions. I would na u-
ally be o in e es o explo e he po en ial use ulness o such
a c i e ion in a ious mo e complex se ings.
ACKNOWLEDGMENTS
We would like o hank Jesu
´sSa
´nchez-Dehesa o he
use ul discussion on he pe u ba ion heo y in he S u m-
Liou ille p oblem. This wo k has been suppo ed by he
Minis e io de Ciencia y Tecnologı
´a o Spain h ough G an
No. BFM2001-3878-C02 and by he Jun a de Andalucı
´a un-
de P ojec No. FQM-0207 共N.R.Q.兲. I has also been pa -
ially suppo ed by NSF unde G an No. DMS-0204585, a
Uni e si y o Massachuse s Facul y Resea ch G an and he
Eppley Founda ion o Resea ch 共P.G.K.兲.
FIG. 5. The oscilla o y beha io o
␻
n
ee⫺
␻
n⫺1
ixed in he nonin-
eg able case (
⑀
⫽0.1) is shown o he i s wa e numbe s 共see he
squa es joined by dashed line兲. The ci cles joined by solid line a e
he esul s o he in eg able sys em (
⑀
⫽0).
FIG. 6. Pe iodic BC o AL-DNLS 共Ablowi z-Ladik DNLS兲o
Eq. 共2兲: The solid line a ze o ep esen s he di e ence be ween wo
consecu i e equencies (n⫽2,4,...) o hein eg able AL la ice
(
⑀
⫽0). The double mul iplici y o he equencies is des oyed as
⑀
is inc eased 共do ed, dashed, and do -dashed lines ep esen he non-
in eg able cases o
⑀
⫽0.1,0.5,1, espec i ely兲.
P. G. KEVREKIDIS AND N. R. QUINTERO PHYSICAL REVIEW E 68, 036612 共2003兲
036612-6
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