Normal form for travelling kinks
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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


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Publié par
Nombre de lectures 9
Langue English

Exrait

†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-
tiondivergingfrommoving

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