Normal form for travelling kinks

†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
fi
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
)
,φ
(
z
)
,φ
(
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
fi
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-
tiondivergingfrommoving