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Normal form for travelling kinks

De
38 pages
Normal form for travelling kinks in discrete Klein–Gordon lattices Gerard Iooss† and Dmitry E. Pelinovsky†† † Institut Universitaire de France, INLN, UMR CNRS-UNSA 6618, 1361 route des Lucioles, 06560 Valbonne, France †† Department of Mathematics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4K1 October 18, 2005 Abstract We study travelling kinks in the spatial discretizations of the nonlinear Klein–Gordon equa- tion, which include the discrete ?4 lattice and the discrete sine–Gordon lattice. The differential advance-delay equation for travelling kinks is reduced to the normal form, a scalar fourth-order differential equation, near the quadruple zero eigenvalue. We show numerically non-existence of monotonic kinks (heteroclinic orbits between adjacent equilibrium points) in the fourth-order equation. Making generic assumptions on the reduced fourth-order equation, we prove the per- sistence of bounded solutions (heteroclinic connections between periodic solutions near adjacent equilibrium points) in the full differential advanced-delay equation with the technique of center manifold reduction. Existence and persistence of multiple kinks in the discrete sine–Gordon equa- tion are discussed in connection to recent numerical results of [ACR03] and results of our normal form analysis. 1 Introduction Spatial discretizations of the nonlinear partial differential equations represent discrete dynamical sys- tems, which are equivalent to chains of coupled nonlinear oscillators or discrete nonlinear lattices.

  • solutions near adjacent

  • travelling kinks

  • no monotonic

  • discrete ?4

  • since discrete

  • periodic solutions near

  • equilibrium state

  • while no

  • zero equilibrium


Voir plus Voir moins

†Normalformfortravellingkinks
indiscreteKlein–Gordonlattices

GerardIooss

andDmitryE.Pelinovsky

InstitutUniversitairedeFrance,INLN,UMRCNRS-UNSA6618,1361routedesLucioles,06560Valbonne,France
DepartmentofMathematics,McMasterUniversity,1280MainStreetWest,Hamilton,Ontario,Canada,L8S4K1
October18,2005

††Abstract
WestudytravellingkinksinthespatialdiscretizationsofthenonlinearKlein–Gordonequa-
tion,whichincludethediscrete
φ
4
latticeandthediscretesine–Gordonlattice.Thedifferential
advance-delayequationfortravellingkinksisreducedtothenormalform,ascalarfourth-order
differentialequation,nearthequadruplezeroeigenvalue.Weshownumericallynon-existence
ofmonotonickinks(heteroclinicorbitsbetweenadjacentequilibriumpoints)inthefourth-order
equation.Makinggenericassumptionsonthereducedfourth-orderequation,weprovetheper-
sistenceofboundedsolutions(heteroclinicconnectionsbetweenperiodicsolutionsnearadjacent
equilibriumpoints)inthefulldifferentialadvanced-delayequationwiththetechniqueofcenter
manifoldreduction.Existenceandpersistenceofmultiplekinksinthediscretesine–Gordonequa-
tionarediscussedinconnectiontorecentnumericalresultsof[ACR03]andresultsofournormal
formanalysis.

1Introduction

Spatialdiscretizationsofthenonlinearpartialdifferentialequationsrepresentdiscretedynamicalsys-
tems,whichareequivalenttochainsofcouplednonlinearoscillatorsordiscretenonlinearlattices.
Motivatedbyvariousphysicalapplicationsandrecentadvancesinmathematicalanalysisofdiscrete
lattices,weconsiderthediscreteKlein–Gordonequationintheform:
u
n
+1

2
u
n
+
u
n

1
u
¨
n
=
2
+
f
(
u
n

1
,u
n
,u
n
+1
)
,
(1.1)
hwhere
u
n
(
t
)

R
,
n

Z
,
t

R
,
h
isthelatticestepsize,and
f
(
u
n

1
,u
n
,u
n
+1
)
isthenonlinearity
function.Thediscretelattice(1.1)isadiscretizationofthecontinuousKlein–Gordonequation,which
emergesinthesingularlimit
h

0
:
u
tt
=
u
xx
+
F
(
u
)
,
(1.2)
1

where
u
(
x,t
)

R
,
x

R
,and
t

R
.Inparticular,westudytwoversionsoftheKlein–Gordon
equation(1.2),namelythe
φ
4
model

2u
tt
=
u
xx
+
u
(1

u
)

()3.1

andthesine–Gordonequation
u
tt
=
u
xx
+sin(
u
)
.
(1.4)
Weassumethatthespatialdiscretization
f
(
u
n

1
,u
n
,u
n
+1
)
ofthenonlinearityfunction
F
(
u
)
issym-
metric,
f
(
u
n

1
,u
n
,u
n
+1
)=
f
(
u
n
+1
,u
n
,u
n

1
)
,
andconsistentwiththecontinuouslimit,

1()5.

f
(
u,u,u
)=
F
(
u
)
.
(1.6)
Weassumethatthezeroequilibriumstatealwaysexistswith
F
(0)=0
and
F

(0)=1
.Thisnormal-
izationallowsustorepresentthenonlinearityfunctionintheform:
f
(
u
n

1
,u
n
,u
n
+1
)=
u
n
+
Q
(
u
n

1
,u
n
,u
n
+1
)
,
(1.7)
wherethelinearpartisuniquelynormalized(parameter
h
canbechosensothatthelineartermof
u
n

1
+
u
n
+1
iscancelled)andthenonlinearpartisrepresentedbythefunction
Q
(
u
n

1
,u
n
,u
n
+1
)
.
Inaddition,weassumethat(i)
F
(
u
)
and
Q
(
u
n

1
,u
n
,u
n
+1
)
areoddsuchthat
F
(

u
)=

F
(
u
)
and
Q
(

v,

u,

w
)=

Q
(
v,u,w
)
and(ii)apairofnon-zeroequilibriumpoints
u
+
=

u


=0
exists,
suchthat
F
(
u
±
)=0
,F

(
u
±
)
<
0
,
(1.8)
whilenootherequilibriumpointsexistintheinterval
u

[
u

,u
+
]
.Forinstance,thisassumptionis
veri

edforthe
φ
4
model(1.3)with
u
±
=
±
1
andforthesine–Gordonequation(1.4)with
u
±
=
±
π
.
WeaddressthefundamentalquestionofexistenceoftravellingwavesolutionsinthediscreteKlein–
Gordonlattice(1.1).SincediscreteequationshavenotranslationalandLorentzinvariance,unlikethe
continuousKlein–Gordonequation(1.2),existenceoftravellingwaves,pulsatingtravellingwavesand
travellingbreathersrepresentsachallengingproblemofappliedmathematics(seerecentreviewsin
[S03,IJ05]).
Ourworkdealswiththetravellingkinksbetweenthenon-zeroequilibriumstates
u
±
.Wearenot
interestedinthetravellingbreathersandpulsatingwavesnearthezeroequilibriumstatesincethezero
stateislinearlyunstableinthedynamicsofthediscreteKlein–Gordonlattice(1.1)with
F

(0)=1
>
0
.Indeed,lookingforsolutionsintheform
u
n
(
t
)=
e
iκhn
+
λn
,wederivethedispersionrelationfor
linearwavesnearthezeroequilibriumstate:
λ
2
=1

4sin
2
κh.
22h2

Itfollowsfromthedispersionrelationthatthereexists
κ

(
h
)
>
0
suchthat
λ
2
>
0
for
0

κ<κ

(
h
)
.
Ontheotherhand,thenon-zeroequilibriumstates
u
+
and
u

areneutrallystableinthedynamicsof
thediscreteKlein–Gordonlattice(1.1)with
F

(
u
±
)
<
0
.Wefocushenceonboundedheteroclinic
orbitswhichconnectthestablenon-zeroequilibriumstates
u

and
u
+
intheform:

u
n
(
t
)=
φ
(
z
)
,z
=
hn

ct,
(1.9)

wherethefunction
φ
(
z
)
solvesthedifferentialadvance-delayequation:
φ
(
z
+
h
)

2
φ
(
z
)+
φ
(
z

h
)
c
2
φ

(
z
)=+
φ
(
z
)+
Q
(
φ
(
z

h
)

(
z
)

(
z
+
h
))
.
(1.10)
2hWeconsiderthefollowingclassofsolutionsofthedifferentialadvance-delayequation(1.10):(i)
φ
(
z
)
istwicecontinuouslydifferentiablefunctionon
z

R
;(ii)
φ
(
z
)
ismonotonicallyincreasingon
z

R
and(iii)
φ
(
z
)
satis

esboundaryconditions:

zlim
φ
(
z
)=
u

,
lim
φ
(
z
)=
u
+
.
(1.11)

→−∞z→+∞ItiseasytoverifythatthecontinuousKlein–Gordonequation(1.2)with
F
(
u
)
in(1.8)yieldsatrav-
ellingkinksolutionintheform
u
=
φ
(
z
)
,
z
=
x

ct
for
|
c
|
<
1
.However,thetravellingkink
canbedestroyedinthediscreteKlein–Gordonlattice(1.1),whichresultsinviolationofoneormore
conditionson
φ
(
z
)
.Forinstance,theboundedtwicecontinuouslydifferentiablesolution
φ
(
z
)
may
developnon-vanishingoscillatorytailsaroundtheequilibriumstates
u
±
[IJ05].
Arecentprogressontravellingkinkswasreportedforthediscrete
φ
4
model.Fourparticularspa-
tialdiscretizationsofthenonlinearity
F
(
u
)=
u
(1

u
2
)
wereproposedwithfourindependentand
alternativemethods[BT97,S97,FZK99,K03],wheretheultimategoalwastoconstructafamily
oftranslation-invariantstationarykinksfor
c
=0
,thataregivenbycontinuous,monotonicallyin-
creasingfunctions
φ
(
z
)
on
z

R
withtheboundaryconditions(1.11).Exceptionaldiscretizations
weregeneralizedin[BOP05,DKY05a,DKY05b],wheremulti-parameterfamiliesofcubicpolyno-
mials
f
(
u
n

1
,u
n
,u
n
+1
)
wereobtained.Itwasobservedinnumericalsimulationsofthediscrete
φ
4
model[S97]thattheeffectivetranslationalinvarianceofstationarykinksimpliesreductionofradia-
tiondivergingfrommovingkinks.Neweffectssuchasself-accelerationwerereportedforsomeofthe
exceptionaldiscretizationsin[DKY05a,DKY05b].Nevertheless,fromamathematicallystrictpoint
ofview,weshallaskifexceptionaldiscretizationsguaranteeexistenceoftruetravellingkinks(hetero-
clinicorbits)atleastforsmallvaluesof
c
.Thequestionwasanswerednegativelyin[OPB05],where
numericalanalysisofbeyond-all-orderexpansionswasdeveloped.Itwasshownthatbifurcationsof
travellingkinksolutionsfrom
h
=0
tosmallnon-zero
h
donotgenerallyoccurinthediscrete
φ
4
modelwithsmallvaluesof
c
,eveniftheexceptionaldiscretizationsallowthesebifurcationsfor
c
=0
.
Itwasalsodiscoveredin[OPB05]thatbifurcationsoftravellingkinksolutionsmayoccurfor

nitely
manyisolatedvaluesof
c
farfromthelimit
c
=0
.

3

Thediscretesine-Gordonmodelwasalsosubjectofrecentstudies.Numericalcomputationswereused
toidentifyoscillatorytailsofsmallamplitudesonthetravellingkinksolutions[EF90,SZE00].These
tailswereexplainedwithanalysisofcentralmanifoldreductions(carriedoutwithoutthenormalform
reductions)[FR05].Exceptionaldiscretizationsofthenonlinearity
F
(
u
)=sin(
u
)
weresuggested
bythetopologicalboundmethodin[SW94](seethereviewin[S99])andbythe”inverse”(direct
substitution)methodin[FZK99].Simultaneouslywiththeabsenceofsinglekinksinthesine-Gordon
lattices,multiplekinks(betweennon-zeroequilibriumpointsof
π
(mod2
π
)
)werediscoveredwiththe
formalreductionofthediscretelattice(1.1)tothefourth-orderODEproblemin[CK00]andcon

rmed
withnumericalanalysisofthedifferentialadvanced-delayequationin[ACR03].
Ourworkismotivatedbytherecentadvancesinstudiesofdifferentialadvanced-delayequationsfrom
thepointofdynamicalsystemmethodssuchascentralmanifoldreductionsandnormalforms(see
pioneerworksin[I00,IK00]).Thesamemethodswererecentlyappliedin[PR05]totravellingsolitary
wavesindiscretenonlinearSchro¨dingerlatticesnearthemaximumgroupvelocityoflinearwaves.It
wasshownin[PR05]thatnon-existenceoftravellingsolitarywavescanbederivedalreadyinthe
truncated(polynomial)normalform.Weshallexploitthisideatotheclassoftravellingkinksin
discreteKlein–Gordonlatticesnearthespeci

cspeed
c
=1
andthecontinuouslimit
h
=0
.This
particularpointcorrespondstothequadruplezeroeigenvalueonthecentralmanifoldofthedynamical
system.Ageneralreversiblenormalformforthequadruplezeroeigenvaluewasderivedandstudied
in[I95](seealsotutorialsinthebook[IA98]).
WenotethatthediscreteKlein–Gordonlatticewasconsideredin[IK00]butthenonlinearity
F
(
u
)
was
takentobedecreasingnear
u
=0
,suchthatthequadruplezeroeigenvaluewasnotobservedinthelist
ofpossiblebifurcationsoftravellingwavesolutions(seeFigure1in[IK00]).Ontheotherhand,the
quadruplezeroeigenvalueoccurredinthediscreteFermi–Pasta–Ulamlatticestudiedin[I00],where
thesymmetrywithrespecttotheshifttransformationwasusedtoreducethebifurcationproblemtothe
three-dimensionalcentermanifold.SincethereversiblesymmetryoperatorfortheFermi-Pasta-Ulum
latticeisminusidentitytimesthereversibilitysymmetryfortheKlein–Gordonlattice,thereversible
normalformsforquadruplezeroeigenvaluearedifferentinthesetwoproblems.Wefocushereonly
onthecaseofthediscreteKlein–Gordonlattice.
Ourstrategyisasfollows.Weshalldevelopadecompositionofsolutionsofthedifferentialadvance-
delayequationintotwoparts:afour-dimensionalprojectiontothesubspaceofthequadruplezero
eigenvalueandanin

nite-dimensionalprojectiontothehyperbolicpartoftheproblem.Byusinga
suitablescaling,weshalltruncatetheresultingsystemofequationswithascalarfourth-orderequation,
whichissimilartotheoneformallyderivedin[CK00].Werefertothisfourth-orderequationas
tothenormalformfortravellingkinks.Weshalldevelopanumericalanalysisofthefourth-order
equationandshowthatnomonotonicheteroclinicorbitsfrom
u

to
u
+
existbothinthediscrete
φ
4
and
sine-Gordonlattices.Rigorouspersistenceanalysisofboundedsolutions(heteroclinicorbitsbetween

4

periodicsolutionsnear
u

and
u
+
)isdevelopedwiththetechniqueofcentermanifoldreduction.Our
mainconclusionisthatthedifferentialadvance-delayequation(1.10)hasnomonotonictravelling
kinksnearthepoint
c
=1
and
h
=0
butitadmitsfamiliesofnon-monotonictravellingkinkswith
oscillatorytails.
Itseemssurprisingthatthetruncatednormalformisindependentonthediscretizationsofthenonlin-
earityfunction
f
(
u
n

1
,u
n
,u
n
+1
)
andthenegativeresultextendstoallexceptionaldiscretizationscon-
structedin[SW94,S97,S99,BT97,FZK99,K03,BOP05,DKY05a,DKY05b].Anyone-parameter
curvesofthemonotonictravellingkinks,whichbifurcateontheplane
(
c,h
)
fromthe

nitesetof
isolatedpoints
(
c

,
0)
(seenumericalresultsin[OPB05]),mayonlyexistfarfromthepoint
(1
,
0)
.
Inaddition,weshallexplainbifurcationsofmultiplekinksfrom
u

=

π
to
u
+
=
π
(2
n

1)
,
n>
1
inthediscretesine-Gordonequationfromthepointofnormalformanalysis.Thesebifurcations
mayoccuralonganin

nitesetofcurvesontheplane
(
c,h
)
whichallintersectthepoint
(1
,
0)
(see
numericalresultsin[ACR03]).Weshallalsoderivethetruncatednormalformfromthediscretizations
oftheinversemethodreportedin[FZK99]andshowthatitmayadmitspecialsolutionsformonotonic
kinks,whenthecontinuitycondition(1.6)isviolated.
Weemphasizethatourmethodsareverydifferentfromthecomputationsofbeyond-all-orderexpan-
sions,exploitedinthecontextofdifferencemapsin[TTJ98,T00]anddifferentialadvanced-delay
equationsin[OPB05].Byworkingneartheparticularpoint
c
=1
and
h
=0
,weavoidbeyond-all-
orderexpansionsandderivenon-existenceresultsatthepolynomialnormalform.Inanasymptotic
limitofcommonvalidity,bifurcationsofheteroclinicorbitsinthetruncatednormalformcanbestud-
iedwithbeyond-all-ordercomputations(seerecentanalysisandreviewin[TP05]).
Ourpaperhasthefollowingstructure.Section2discusseseigenvaluesofthelinearizationproblem
atthezeroequilibriumpointandgivesaformalderivationofthemainresult,thescalarfourth-order
equation.Therigorousderivationofthescalarfourth-orderequationfromdecompositionsofsolutions,
projectiontechniquesandtruncationisdescribedinSection3.Section4containsnumericalanalysis
ofthefourth-orderequation,wherewecomputethesplitfunctionforheteroclinicorbits.Persistence
analysisofboundedsolutionsinthedifferentialadvance-delayequationisdevelopedinSection5.
Section6concludesthepaperwithsummaryandopenproblems.Anumberofimportantbuttechnical
computationsarereportedinappendicestothispaper.AppendixAcontainsthecomparisonofthe
fourth-orderequationwiththenormalformfrom[I95].AppendixBgivestheproofofexistenceof
centermanifoldinthefullsystem,whichsupplementstheanalysisin[IK00].AppendixCdescribes
computationsoftheStokesconstantforheteroclinicorbitsinthefourth-orderdifferentialequation
frommethodsof[TP05].AppendixDdiscussesapplicationsofthefourth-orderequationtotheinverse
methodfrom[FZK99].

5

2Resonancesindispersionrelationsandthescalarnormalform

Whenthedifferentialadvance-delayequation(1.10)islinearizednearthezeroequilibriumpoint,we
lookforsolutionsintheform
φ
(
z
)=
e
λz
,where
λ
belongstothesetofeigenvalues.Alleigenvalues
ofthelinearizedproblemsareobtainedfromrootsofthedispersionrelation:
D
(Λ;
c,h
)=2(coshΛ

1)+
h
2

c
2
Λ
2
=0
,
(2.1)
where
Λ=
λh
istheeigenvalueinzoomedvariable
ζ
=
hz
.Weareinterestedtoknowhowmany
eigenvaluesoccurontheimaginaryaxisandwhethertheimaginaryaxisisisolatedfromthesetof
complexeigenvalues(thatiscomplexeigenvaluesdonotaccumulateattheimaginaryaxis).Imaginary
eigenvalues
Λ=2
iK
ofthedispersionrelation(2.1)satisfythetranscendentalequation:
2sin
2
K
=
h
+
c
2
K
2
.
(2.2)
4Particularresultsonrootsofthetranscendentalequation(2.2)areeasilydeducedfromanalysisof
thefunction
sin(
x
)
incomplexdomain.Duetotheobvioussymmetry,weshallonlyconsiderthe
non-negativevaluesof
c
and
h
.

••When
c
=0
and
0
<h<
2
,equation(2.2)hasonlysimplerealroots
K
,suchthatalleigenval-
ues
Λ
arepurelyimaginaryandsimple.Allrealroots
K
becomedoubleat
h
=0
and
h
=2
.
When
h
=0
and
c

=1
,adoublezerorootof
K
(adoublezeroeigenvalue
Λ
)exists.When
0
<c<
1
,

nitelymanypurelyimaginaryeigenvalues
Λ
exist(e.g.onlyonepairofroots
K
=
±
P
existsfor
c
0
<c<
1
,where
c
0

0
.
22
),whilein

nitelymanyrootsarecomplex
anddistantawaytheimaginaryaxis.When
c>
1
,allnon-zeroeigenvalues
Λ
arecomplexand
distantawaytheimaginaryaxis.

Inthegeneralcaseof
h

=0
and
c

=0
,thetranscendentalequation(2.2)ismorecomplicatedbutit
canbeanalyzedsimilarlytothedispersionrelationconsideredin[I00,IK00].

Lemma2.1
Thereexistsacurve
h
=
h

(
c
)
,whichintersectsthepoints
(1
,
0)
and
(0
,
2)
ontheplane
(
c,h
)
,suchthatfor
0
<c<
1
and
0
<h<h

(
c
)

nitelymanyeigenvalues
Λ
ofthedispersion
relation(2.1)arelocatedontheimaginaryaxisandallothereigenvaluesareinacomplexplanedistant
fromtheimaginaryaxis.Thecurve
h
=
h

(
c
)
correspondstothe1:1resonanceHopfbifurcation,
wherethesetofimaginaryeigenvaluesincludesonlyonepairofdoubleeigenvalues.

Remark2.1
The1:1resonantHopfbifurcationisillustratedonFigure1.Figure1(a)showsagraph-
icalsolutionofthetranscendentalequation(2.2)for
h
=
h

(
c
)
.Figure1(b)showsthebifurcation
curve
h
=
h

(
c
)
ontheplane
(
c,h
)
.Thecurve
h
=0
for
0
<c<
1
correspondstothedoublezero
eigenvalueinresonancewithpairsofsimplepurelyimaginaryeigenvalues.

6

Proof.
Let
Λ=
p
+
iq
andrewritethedispersionrelationintheequivalentform:

c
2
(
q
2

p
2
)+
h
2
+2(cosh
p
cos
q

1)=0
c
2
pq

sinh
p
sin
q
=0
.

Itfollowsfromthesystemwith
c

=0
thattheimaginarypartsoftheeigenvaluesisboundedbythe
realpartsoftheeigenvalues:
q
2

p
2
+4
2
cosh
2
p.
(2.3)
2cTherefore,ifcomplexeigenvaluesaccumulatetotheimaginaryaxis,suchthatthereexistsasequence
(
p
n
,q
n
)
with
lim
n
→∞
p
n
=0
and
lim
n
→∞
q
n
=
q

,theaccumulationpoint
(0
,q

)
isbounded.
However,since
D
(Λ;
c,h
)
isanalyticin
Λ
,thedispersionrelation(2.1)mayhave

nitelymanyroots
of

nitemultiplicitiesinaboundeddomainofthecomplexplane.Therefore,theaccumulationpoint
(0
,q

)
doesnotexistandcomplexeigenvaluesaredistantfromtheimaginaryaxis.Bythesamereason,
thereexist

nitelymanyeigenvaluesontheimaginaryaxis
Λ

i
R
.
Lookingatthedoublerootsat
K
=
±
P
,we

ndaparametrizationofthecurve
h
=
h

(
c
)
intheform:
sin
P
cos
P
c
2
=
,h
2
=4sin
P
(sin
P

P
cos
P
)
.
(2.4)
PAsimplegraphicalanalysisofthetranscendentalequation(2.2)showsthatthedoublerootsat
K
=
±
P
areuniquefor
0
<P<
2
π
,whentheintersectionoftheparabolaandthetrigonometricfunction
occursatthe

rstfundamentalperiodof
sin
2
-function.At
P
=0
(atthepoint
(
c,h
)=(1
,
0)
),thepair
ofdoubleimaginaryeigenvalues
Λ=
±
2
iP
mergeat
Λ=0
andformaquadruplezeroeigenvalue.
πAt
P
=
2
(atthepoint
(
c,h
)=(0
,
2)
),asequenceofin

nitelymanydoubleimaginaryeigenvalues
existsat
Λ=

(1+2
n
)
,
n

Z
.

Wewillbeinterestedinthereductionofthedifferentialadvanced–delayequation(1.10)atthepartic-
ularpoint
(
c,h
)=(1
,
0)
.Let

beasmallparameterthatde

nesapointontheplane
(
c,h
)
,whichis
locallyclosetothebifurcationpoint
(1
,
0)
:

c
2
=1+
γ,h
2
=

2
τ.
(2.5)

√Let
Λ=

Λ
1
bethescaledeigenvaluenear
Λ=0
,sothattheTaylorseriesexpansionofthedisper-
sionrelation(2.1)allowsforanon-trivialbalanceof

-terms.TruncatingtheTaylorseriesbeyondthe
leadingorder
O(

2
)
,weobtainthebi-quadraticequationforthescaledeigenvalue
Λ
1
:

14

γ
Λ
12
+
τ
=0
.
(2.6)
21The1:1resonanceHopfbifurcationcorrespondstothepoint,wheretherootsofthebi-quadraticequa-
tion(2.6)aredoubleandpurelyimaginary.Thispointoccurswhen
τ
=3
γ
2
,whichagreeswiththe

7

5.10.25h
2
+c
2
K
2

15.0

010 P

2h5.11sin
2
(K)

5.0

h = h
*
(c)

h = 0
02
K
300.20.40.60.8
c
1

πFigure1:Left:Thegraphicalsolutionofthetranscendentalequation(2.2)for
P
=
4
,where
(
c,h
)
are
givenby(2.4).Right:Thebifurcationcurvefor1:1resonantHopfbifurcation
h
=
h

(
c
)
anddouble
zeroresonancebifurcation
h
=0
.
Taylorseriesexpansionofthesystem(2.4)inpowersof
P
attheleadingordersof
P
2
and
P
4
.Asa
result,thebifurcationcurve
h
=
h

(
c
)
hastheleadingorderbehavior:
√h

(
c
)=3(1

c
2
)+o(1

c
2
)
.
(2.7)
Thenormalformfortravellingkinks,whichisthemainresultofourpaper,canberecoveredwitha
formalasymptoticexpansionofthenonlineardifferentialadvance-delayequation(1.10).Let
ζ
=
hz
andrewritethemainequation(1.10)intheform:
c
2
φ

(
ζ
)=
φ
(
ζ
+1)

2
φ
(
ζ
)+
φ
(
ζ

1)+
h
2
φ
(
ζ
)+
h
2
Q
(
φ
(
ζ

1)

(
ζ
)

(
ζ
+1))
.
(2.8)
Let
(
c,h
)
betheperturbedbifurcationpointintheform(2.5).Let
φ
(
ζ
)
beasmoothfunctionofa
√√√slowlyvaryingvariable
ζ
1
=
ζ
.Expanding
φ
(
ζ
1
±

)
intheTaylorseriesin

andtruncating
bytermsbeyondtheleadingorder

2
,weobtainthescalarfourth-orderdifferentialequation:

(iv)
(
ζ
1
)

γφ

(
ζ
1
)+
τF
(
φ
(
ζ
1
))=0
,
(2.9)
21wherewehaveusedtherepresentations(1.6)and(1.7).ThelinearizationofthenonlinearODE
(2.9)nearthezeroequilibriumpointrecoversthebi-quadraticdispersionrelation(2.6).Thenon-
linearequation(2.9)hastheequilibriumpoints
u

,
0
,and
u
+
,whichareinheritedfromtheequilib-
riumpointsoftheKlein–Gordonequation(1.2).Ourmainexampleswillincludethe
φ
4
modelwith
F
(
u
)=
u
(1

u
2
)
,
u
±
=
±
1
andthesine–Gordonequationwith
F
(
u
)=sin(
u
)
,
u
±
=
±
π
.

8

3Decompositions,projectionsandtruncation

Weshallderivethenormalformequation(2.9)withrigorousanalysiswhenthesolutionofthedif-
ferentialadvance-delayequation(2.8)isdecomposednearaquadruplezeroeigenvalue.Weadopt
notationsof[IK00](seereviewin[IJ05])andrewritethedifferentialadvance-delayequationasan
in

nite-dimensionalevolutionproblem.Weshallworkwiththescaled(inner)variable
ζ
=
hz
,where
thedifferentialadvance-delayequation(1.10)takestheform(2.8).Let
p
beanewindependentvari-
able,suchthat
p

[

1
,
1]
andde

nethevector
U
=(
U
1
(
ζ
)
,U
2
(
ζ
)
,U
3
(
ζ,p
))
T
,suchthat
U
1
=
φ
(
ζ
)
,U
2
=
φ

(
ζ
)
,U
3
=
φ
(
ζ
+
p
)
.
(3.1)
Itisclearthat
U
3
(
ζ,p
)=
U
1
(
ζ
+
p
)
and
U
3
(
ζ,
0)=
U
1
(
ζ
)
.Thedifferenceoperatorsarethende

ned
saU
3
(
ζ,
±
1)=
δ
±
U
3
(
ζ,p
)=
φ
(
ζ
±
1)=
δ
±
U
1
(
ζ
)=
U
1
(
ζ
±
1)
.
(3.2)
Let
D
and
H
bethefollowingBanachspacesfor
U
=(
U
1
,U
2
,U
3
(
p
))
T
,
=(
U
1
,U
2
)

R
2
,U
3

C
1
([

1
,
1]
,
R
)
,U
3
(0)=
U
1
,
(3.3)
=(
U
1
,U
2
)

R
2
,U
3

C
0
([

1
,
1]
,
R
)
,
(3.4)
withtheusualsupremumnorm.Welookforasmoothmapping
ζ

U
(
ζ
)
in
C
0
(
R
;
D
)
,which
representsclassicalsolutionsofthein

nite-dimensionalevolutionproblem:
2hdU
=
L
c,h
U
+
2
M
0
(
U
)
,
(3.5)
cζdwhere
L
c,h
and
M
0
(
U
)
arefoundfromthedifferentialadvance-delayequation(2.8):
⎛010⎜=

h
22

2
0
1
2
(
δ
+
+
δ

)
(3.6)
⎝cc00∂

DHLcM,0hp⎞⎟⎟⎠dna(
U
)=(0
,Q
(
δ

U
3
,U
1

+
U
3
)
,
0)
T
.
(3.7)
Thelinearoperator
L
c,h
maps
D
into
H
continuouslyandithasacompactresolventin
H
.The
nonlinearity
M
0
(
U
)
isanalyticinanopenneighborhoodof
U
=
0
,maps
H
into
D
continuously,and


M
0
(
U
)

D
=O

U

2
H
.
Thespectrumof
L
c,h
consistsofanin

nitesetofisolatedeigenvaluesof

nitemultiplicities.By
virtueoftheLaplacetransform,eigenvaluesofthelinearoperator
L
c,h
arefoundwiththesolution
(
ζ,p
)=(1
,
Λ
,e
Λ
p
)
T
e
Λ
ζ
,whenthelinearproblem
U

(
ζ
)=
L
c,h
U
isreducedtothedispersion

U9

L1,0Ujrelation(2.1),thatis
D
(Λ;
c,h
)=0
.Weareparticularlyinterestedinthebifurcationpoint
c
=1
and
h
=0
,when
D
1
(Λ)

D
(Λ;1
,
0)=2(coshΛ

1)

Λ
2
.
(3.8)
Thetranscendentalequation
D
1
(Λ)=0
hasthequadruplezerorootandnootherrootintheneigh-
borhoodof
Λ

i
R
.ThefourgeneralizedeigenvectorsoftheJordanchainforthezeroeigenvalue,
=
U
j

1
,
j
=1
,
2
,
3
,
4
with
U

1
=
0
,arefoundexactlyas:
⎛⎞⎛⎞⎛⎞⎛⎞

1
⎟⎜
0
⎟⎜
0
⎟⎜
0

=
⎝⎜
0
⎠⎟
,
U
1
=
⎝⎜
1
⎠⎟
,
U
2
=
⎝⎜
0
⎠⎟
,
U
3
=
⎝⎜
0

.
(3.9)
1
p
21
p
21
p
Thedynamicalsystem(3.5)hasthereversibilitysymmetry
S
,suchthat
2hdS
U
=
L
c,h
S
U
+
2
M
0
(
S
U
)
,
cζd

U0−SSUj63⎠(3.10)

erehwTU
=(
U
1
(
ζ
)
,

U
2
(
ζ
)
,U
3
(
ζ,

p
))
(3.11)
andthesymmetryproperty(1.5)hasbeenused.Sinceboththelinearoperator(3.6)andthenonlinearity
(3.7)anti-commutewiththereversibilityoperator(3.11),thestandardpropertyofreversiblesystems
holds:if
U
(
ζ
)
isasolutionof(3.5)forforwardtime
ζ>
0
,then
S
U
(

ζ
)
isasolutionof(3.5)for
backwardtime
ζ<
0
(see[LR98]forreview).Applyingthereversibilityoperator
S
totheeigenvectors
(3.9)atthebifurcationofthequadruplezeroeigenvalue,weobservethat
=(

1)
j
U
j
,j
=0
,
1
,
2
,
3
.
(3.12)
Thisbifurcationcase

tstotheanalysisof[I95],wherethereversiblenormalformwasderivedforthe
quadruplezeroeigenvalue.AppendixAshowsthatthegeneralreversiblenormalformfrom[I95]is
reducedtothenormalformequation(2.9)byappropriatescaling.However,wenoticethatthisresult
cannotbeapplieddirectly,sinceitisonlyvalidinaneighborhoodoftheorigin,whileinthepresent
case,thesolutionsweareinterestedinareoforderof
O(1)
.
Inordertostudybifurcationofquadruplezeroeigenvalueinthereversiblesystem(3.5),weshallde

ne
theperturbedpoint
(
c,h
)
nearthebifurcationpoint
(1
,
0)
withanexplicitsmallparameter
ε
.Contrary
tothede

nition(2.5),itwillbeeasiertoworkwiththeparameters
(
γ,τ
)
,de

nedfromtherelations:
21h22
=1

εγ,
2
=
ετ,
(3.13)
ccwhere
ε
isdifferentfrom

usedin(2.5).Thedynamicalsystem(3.5)withtheparametrization(3.13)
isrewrittenintheexplicitform:
dU
=
L
1
,
0
U
+
εγ
L
1
(
U
)+
ε
2
τ
L
2
(
U
)+
ε
2
τ
M
0
(
U
)
,
ζd01

(3.14)