Unimodal wave trains and solitons in convex FPU chains

0.46 MB·English

by  Michael Herrmann

Unimodal wave trains and solitons in convex FPU chains
∗
Michael Herrmann
January 23, 2009
9
0
0 Abstract
2
Weconsideratomicchainswithnearestneighbourinteractionsandstudyperiodicandhomoclinic
n
travelling waves which are called wave trains and solitons, respectively. Our main result is a new
a
existenceproofwhichreliesontheconstrainedmaximisationofthepotentialenergyandexploitsthe
J
invariance properties of an improvement operator. The approach is restricted to convex interaction
3
potentials but refines the standard results as it provides the existence of travelling waves with uni-
2
modal and even profile functions. Moreover, we discuss the numerical approximation and complete

] localizationofwavetrains,andshowthatwavetrainsconvergetosolitonswhentheperiodicitylength
h tendstoinfinity.
p
-
h Keywords: Fermi-Pasta-Ulam chain, lattice travelling waves,
t constrained optimisation, complete localisation
a
m MSC (2000): 37K60, 47J30, 70F45, 74J30
[
Contents

1
v 1 Introduction 1
6
3 2 Variationalapproach 4
2.1 Travellingwavesaseigenfunctionsofnonlinearintegralequations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
7
2.2 Variationalstructureforwavetrainsandsolitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 2.3 Somepiecesoffunctionalanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
.
1 3 WaveTrains 11
0 3.1 Existenceresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9 3.2 Completelocalisationofwavetrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
0
4 Solitons 16
:
v 4.1 Existenceofsolitonsforgenuinelysuper-quadraticP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Criterionsforsuper-quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
i P
X 4.3 Criterionsforgenuinelysuper-quadraticP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 Solitonsaslimitsofwavetrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
r
a
1 Introduction
We consider infinite chains of identical atoms with unit mass that are coupled by nearest neighbour
interactions. The dynamics in such chains is governed by Newton’s equations
x¨j(t)=Φ′ xj+1(t) xj(t) Φ′ xj(t) xj 1(t) , (1)
− − − −
where x (t) denotes the position of th(cid:0)e jth atom at ti(cid:1)me t,(cid:0)and Φ is the inte(cid:1)ractionpotential. Restating
j
(1) in terms of atomic distances r (t)=x (t) x (t) and atomic velocities v (t)=x˙ (t) we find
j j+1 j j j
−
r˙j(t)=vj+1(t) vj(t), v˙j(t)=Φ′ rj(t) Φ′ rj 1(t) . (2)
− − −
∗UniversityofOxford,MathematicalInstitute, CentreforNonlinear(cid:0)PDE(cid:1)(OxPD(cid:0)E),24-29(cid:1)StGiles’,OxfordOX13LB,
UnitedKingdom,[email protected].
1
InthisarticleweallowforarbitraryconvexinteractionpotentialsΦandreferto(1)asFermi-Pasta-Ulam
(FPU) chain although the potential in the original paper [FPU55] was a quartic polynomial.
FPU chains can be viewed as simple toy models for crystals and solids and allow to study some
essential properties of nonlinear elastic materials. Even though (1) is a strong simplification of a real
material it obeys a very complex behaviour and currently we are far from a complete understanding of
its dynamical properties.
Duringthe lastdecades,alotofresearchaddressedthe existenceandpropertiesoftravellingwavesin
FPU chains because they can be viewed as elementary wavesand providea lot of insightinto the energy
transport in nonlinear media. Such travelling waves solve nonlinear advance-delay differential equations
and it is a fundamental mathematical problem to characterise the solution set to those equations.
A further motivation for the study of travelling waves is related to atomistic Riemann problems and
self-thermalisation of FPU chains: Starting with piecewise constant initial data for the atomic distances
and velocities, solutions to (2) are self-similar on a macroscopic scale and involve dispersive shocks, that
are fan-like structures with strong microscopic oscillations, see Figure 1. It is known from the theory
of integrable systems and numerical simulations that the oscillations within a dispersive shock can be
described by modulated travelling waves, compare [FV99, DH08] and references therein.
distributionfunctions
+0.83
y
cit
o
el
v
-0.04
+0.95 distance +2.05
Figure 1: The atoms ina dispersiveshock self-organizeinto modulatedtravellingwaves. Left. Snapshots
ofatomicdistancesandvelocitiesagainstthescaledparticleindexfordifferentmacroscopictimes. Right.
Corresponding family of wave trains with super-sonic soliton as ‘homoclinic limit’; the picture shows
the density plots of six local distribution functions in the (r, v)-plane. Modulation theory predicts that
each of these local distribution functions corresponds to a travelling wave whose parameters depend on
macroscopic time and particle index.
Travelling waves andmainresult Travellingwaves areexactsolutionsto(2)andsatisfytheansatz1
r (t)=R(ϕ), v (t)=V(ϕ), ϕ=kj+ωt, (3)
j j
with phase ϕ, wave number k, (negative) frequency ω, and profile functions R and V. Inserting (3) into
(2) we obtain the nonlinear advance-delay differential equations
d d
ω dϕR=∇+kV, ω dϕV =∇−kΦ′(R), (4)
where ∇+k and ∇−k denote the forward and backward difference operators with shift k, respectively.
Depending on the properties of the profile functions we distinguish the following cases:
1Theansatz(3)isslightlymoregeneral thantheusualone,whichassumesthattheatomicpositionsxj dependonthe
phase variable ϕ. In fact, the ansatz xj(t) = X(kj+ωt) is not invariant under Galilean transformations x x+v0t,
whereas(3)hasthispropertyasitrespectsr r,v v+v0.
2
1. Wave trains or periodic waves: R and V are periodic,
2. Solitons or homoclinic waves: R and V are localized over a constant backgroundstate,
3. Fronts or heteroclinic waves: R and V connect different constant backgroundstates,
4. Oscillatory fronts: R and V connect different asymptotic wave trains.
Notice that our usage of ’soliton’ is quite sloppy: Localized travelling waves are sometimes called ‘soli-
tary waves’, and ‘soliton’ then refers to a solitary waves that survives collisions with other such waves
unchanged.
In this article we show the existence of wave trains and solitons with unimodal and even profile
functions R and V, where even means as usual invariance under ϕ ϕ, and unimodal functions are
−
monotone for both ϕ 0 and ϕ 0. Our main result can be stated as follows.
≤ ≥
Theorem 1.1. Under natural regularity assumptions on the convex potential Φ there exists a four-
parameter family of wave trains, and if Φ satisfies additionally some super-quadratic grow conditions
then there exists also a three-parameter family of solitons. Moreover, the profile functions R and V for
both families are unimodal and even.
Close to our work are [FW94, SW97, FV99, PP00, SZ07] where the existence of travelling waves is
likewise studied in a variational framework. Therefore we shall compare both our method ( 2.2) and
§
results ( 4.3) with those presented therein. We also refer to the numerical study [DEFW93], to [FM03]
§
forexistenceresultsin2Dlattices,andto[Ioo00]whichprovestheexistenceofsmall-amplitudetravelling
waves by means of center-manifold reduction. Moreover, the existence of fronts is studied in [HR08a],
and [SZ08] concerns oscillatory fronts in FPU chains with bi-harmonic potentials.
All results presented below concern solely wave trains and solitons in FPU chains, but the method
can also be applied to other Hamiltonian lattices with convex potential energy as for instance Klein-
P
Gordonchains with convexon-sitepotential, see [Her09], and atomic chainswith next-nearestneighbour
interactions.
We emphasize that we are not able to provide uniqueness results for travelling waves. Uniqueness of
relativeequilibria inHamiltonianlattices is a notoriouslydifficult problem,andalmostnothing is known
about it. The only available results concern either the near-sonic limit [FP99], or systems where the
travelling wave equation can be solved explicitly. Examples are the Toda chain [Tod81], the discrete
nonlinear Schr¨odinger equation [HLM94], the harmonic chain, and the hard-sphere model [DHM06].
Overview on the proofand organisationof the paper Inafirststepwereformulatethetravelling
wave equation (4) in terms of a normalised profile function W L2. More precisely, we show that (4)
∈
can be transformed into a nonlinear eigenvalue equation
ω2W =∂ (W), (5)
P
where (W)isthepotentialenergyofatravellingwave. TheprofilefunctionW hasnophysicalmeaning,
P
ϕ+k
but determines R and V via R(ϕ) =r + W(ϕ˜)dϕ˜ and V(ϕ) =v +ωW(ϕ), where r and v are
0 ϕ 0 0 0
suitable normalization constants.
R
Ourapproachrelies ona combinationof variationalanddynamicalconcepts,andcanbe summarized
as follows.
1. Equation(5) is the Euler-Lagrangeequationfor the optimisationproblem (W) max subjected
to the constraint W , where γ is a free parameter, L2 denotes tPhe bal→l of radius √2γ ,
γ γ
∈ B B ⊂
and ω2 is the Lagrangianmultiplier.
2. There exists an improvement dynamics W [W] on that increases the potential energy.
γ γ
7→ T B
Moreover,each stationary point of this dynamics solves (5), and vise versa.
3
3. There existnon-trivialcones S that are invariantunder the improvementdynamics. Consequently,
each maximiser of in is a travelling wave,see Theorem 2.3.
γ
P S∩B
We emphasize thatthe convexityofΦ is essentialforthis approachas itis intimately relatedto boththe
properties of and the existence of non-trivial invariant cones.
γ
T
A major part of the mathematical analysis done in this paper is needed to show that there exists
a maximizer of in . In the wave-train setting we can use rather simple compactness arguments as
γ
the functional Pis coSntinuous with respect to the weak topology in L2. In the soliton setting, however,
P
we lack the weak compactness of , and the existence proof for maximizers requires more sophisticated
P
arguments. Our main technical result in this context is Lemma 4.7, which implies that (for certain S)
the maximising sequences for in are localized, and hence pre-compact in the strong topology.
γ
P S
This paper is organised as follows. In 2.1 we derive the fixed point equation (5) for both wave
§
trains and solitons. The details of our variational approach are presented in 2.2, and 2.3 concerns the
§ §
analyticalproperties of the underlying functionals and operators. In 3.1 we continue with the existence
§
proof for wave trains and present some numerical simulations; the complete localisation of wave trains
is studied in 3.2. The existence proof for solitons is contained in 4.1 and relies on a natural condition
§ §
for the super-quadratic growth of the functional . In 4.2 and 4.3 we then discuss the corresponding
P § §
properties for the interaction potential Φ. Finally, inspired by the notion of Γ-convergence we show in
4.4 that wave trains converge to solitons when the periodicity length tends to infinity.
§
2 Variational approach
In this section we transform the travelling wave equation into a fixed point equation for a normalised
profile function W, and describe our variational approach to existence results for both wave trains and
solitons. To point our the key ideas we start with more formal consideration in 2.1 and 2.2, and
§ §
postpone the analytical details to 2.3.
§
2.1 Travelling waves as eigenfunctions of nonlinear integral equations
In what follows we assume that the periodicity length of wave trains is given by 2L with 0 < L < ,
∞
andregardthe correspondingprofilefunctionsRandV as definedon[ L, L]. Moreover,weidentify the
−
solitoncasewithL= byconsideringRandV asfunctionson[ , ],or,equivalently,asfunctionson
theAlexandrovcompa∞ctificationofR. Inotherwords,inbothca−se∞sw∞eimposethe’boundaryconditions’
R(L)=R( L) and V(L)=V( L).
Inwhat−followswedenote by−L2([ L, L])with L (0, ]the Lebesguespaceofallsquare-integrable
functions on [ L, L], and if there is n−o risk of confus∈ion w∞e write L2 instead of L2([ L, L]).
− −
Ourfirstaimistotransformthetravellingwaveequationsforwavetrainsandsolitonsintoeigenvalue
equations for certain nonlinear integral operators defined on L2. For this purpose we define two linear
averaging operators, and normalise the potential Φ. More precisely, for given reference distance r we
0
define the potential Φ by
r0
Φ (r)=Φ(r +r) Φ(r ) Φ(r )r
r0 0 − 0 − ′ 0
which is normalised via Φ (0)=Φ (0)=0 and Φ (0)= Φ (r ), and mention that this normalisation
r0 ′r0 ′r′0 ′′ 0
respects the convexity of Φ. The eigenvalue equations for solitons and wave trains involve the averaging
operators
ϕ+k/2
¯ W (ϕ)= W(ϕ˜)dϕ˜,
k
A
Z
(cid:0) (cid:1) ϕ k/2
−
4
and
L L
1 k
A W = ¯ W ¯ W(ϕ˜)dϕ˜= ¯ W W(ϕ˜)dϕ˜.
k k k k
A − 2L A A − 2L
Z Z
L L
b − −
Notice that the operator ¯ is well defined and symmetric on L2 for both finite and infinite L, compare
k
A
Lemma 2.5 below, whereas is well defined for L< only.
k
A ∞
Wave trains and normalbisation via mean values In order to reformulate the travelling wave
equation (4) for L< we introduce the mean values of wave trains by
∞
L L
1 1
r := R(ϕ)dϕ, v := V(ϕ)dϕ.
av av
2L 2L
Z Z
L L
− −
Observation 2.1. With the identification
R(ϕ k/2)=r +( W)(ϕ), V(ϕ)=v +ωW(ϕ), (6)
av k av
− A
for some profile function W with 21L LLW(ϕb)dϕ=0 the integral equation
−
R ω2W = Φ ( W). (7)
Ak ′rav Ak
is equivalent to the wave train equation (4).
b b
Proof. First suppose that R and V solve (4), and let W =ω 1(V v ). Then, the first equation in (4)
− av
−
implies
d d
R(ϕ k/2)=W(ϕ+k/2) W(ϕ k/2)= W (ϕ),
k
dϕ − − − dϕ A
(cid:16) (cid:17)
and hence R(ϕ k/2)=( W)(ϕ)+c for some constantc . Integratingbthis with respect to ϕ we find
k 1 1
− A
r =0+c , and hence (6) . Moreover,the second equation in (4) provides
av 1 1
b
d
ω2 W(ϕ)=Φ′ rav+( kW)(ϕ+k/2) Φ′ rav+( kW)(ϕ k/2)
dϕ A − A −
(cid:16) (cid:17) (cid:16) (cid:17)
=Φ′rav (AkWb)(ϕ+k/2) −Φ′rav (AkW)(ϕb−k/2) ,
(cid:16) (cid:17) (cid:16) (cid:17)
and integration with respect to ϕ givebs ω2W = Φ ( W)+bc for some constant c . The condition
Ak ′rav Ak 2 2
1 L W(ϕ)dϕ=0nowimpliesc =0andhence(6) . NowletasolutionW of (7)begiven,anddefine
2L L 2 2
both−RandV by (6). Then,thefirstequationfrobm(4)hobldsbyconstruction,andthesecondonefollows
R
from (7) by differentiation with respect to ϕ.
Solitonsand normalisationvia background states Forsolitonsitismuchmoreconvenienttobase
the normalisationon the background states
r := lim R(ϕ), v := lim V(ϕ),
bg bg
ϕ ϕ
→±∞ →±∞
because then the normalised profiles are localised.
Observation 2.2. With the identification
R(ϕ k/2)=r +( ¯ W)(ϕ), V(ϕ)=v +ωW(ϕ), (8)
bg k bg
− A
for some profile function W with lim W(ϕ)=0 the integral equation
ϕ
→±∞
ω2W = ¯ Φ ( ¯ W). (9)
Ak ′rbg Ak
is equivalent to the soliton equation (4).
5
Proof. Similar to that of Observation 2.1.
Wecanconsidertheintegralequation(9)alsoforL< ,andreadilyverifythateachsolutiondefines
∞
awavetrainvia(8). ThisnormalisationforwavetrainsseemstobeartificialbecauseforfiniteLwehave
noimmediateinterpretationoftheparameterr . However,in 4.4werelyonthissettingandshowthat
bg
§
the L-periodic solutions W to (9) converge to solitons as L , see Corollary 4.21, an this implies
L
→ ∞
R ( L) r .
L bg
± →
Parameter dependenceoftravelling waves Thetravellingwaveequation(4)obeysasimplescaling
symmetry. In fact, the solution set to (4) is invariant under
R(ϕ) R(λϕ), V(ϕ) V(λϕ), k λ 1k, ω λ 1ω, L λ 1L, (10)
− − −
with arbitraryλ>0, corresponding to
W(ϕ) λW(λϕ) with r r , v v , r r , v v .
bg bg bg bg av av av av
Up to this scaling there remain four independent parameters for wave trains. A natural choice for
nonlinear potentials is to fix the length parameter L, and to regard r , v , k, and ω as independent
av av
parameters,see[DHM06]. Inthispaper,however,weprefertofixthewavenumberk,sothatwavetrains
are parametrised by r , v , L, and the phase speed σ :=ω/k. Accordingly, we parametrize solitons by
av av
r , v and σ.
bg bg
Notice that the velocity parameters v and v do not appear in the travelling wave equations
av bg
due to the Galilean invariance of FPU chains. Both v and v are thus trivial parameters, which
av bg
however become important when studying modulated travelling waves for which the parameters vary on
a macroscopic scale, compare [DHM06, DH08].
2.2 Variational structure for wave trains and solitons
Inviewofthescalingandreformulationresultsfrom 2.1westarttosimplifyoursetting. Inwhatfollows
§
we consider an arbitrary convex and smooth potential Φ normalised by Φ(0)=Φ(0)=0. Moreover,we
′
restrictto k =1, so that the frequency ω equals the phase speed σ, andconsider solely the followingtwo
averagingoperators
ϕ+1/2 L
1
¯W (ϕ)= W(ϕ˜)dϕ˜, (AW)(ϕ)= ¯W (ϕ) W(ϕ)dϕ. (11)
A A − 2L
Z Z
(cid:0) (cid:1) ϕ 1/2 (cid:0) (cid:1) L
− b −
Moreover,inordertopointoutthesimilaritiesbetweenwavetrainsandsolitonswerefertoanabstract
averagingoperator , which equals either ¯or , and consider the general travelling wave equation
A A A
σ2W =b Φ′( W), (12)
A A
with W L2([ L, L]), where L might be finite or infinite. In what follows we call W a travelling wave
∈ −
if and only if there exist σ2 >0 such that (12) is satisfied.
Startingpointforthe variationalformulationisthe observationthat(12)isthe Euler-Lagrangeequa-
tion to the action functional
σ2, W = σ2, W (W),
L K −P
where (cid:0) (cid:1) (cid:0) (cid:1)
L L
σ2, W = 1σ2 W(ϕ)2dϕ and = Φ ( W)(ϕ) dϕ (13)
K 2 P A
Z Z
(cid:0) (cid:1) L L (cid:0) (cid:1)
− −
6
are the kinetic and potential energy, respectively.
The first rigorousexistence resultfor wavetrains and solitonsunder super-quadraticgrowthassump-
tions for Φ was given by Friesecke and Wattis [FW94]. The key idea is (in our notations) to minimise
the L2-normofW under the constraintofprescribedpotentialenergy,where1/σ2 playsthe role ofa La-
grangian multiplier. The existence of corresponding minimisers was then established by means of Lions’
concentration-compactnessprinciple [Lio84].
Smets and Willem [SW97] prove the existence of solitons by showing that for super-quadratic Φ the
functional satisfies the assumptions of (a modified) Mountain Pass Theorem. Recently, these results
L
couldbe improvedbySchwetlickandZimmer. In [SZ07] they requirethe super-quadraticgrowthto hold
only asymptotically, so certain double-well potentials are admissible.
Similarly to [SW97], Pankov and Pflu¨ger [PP00] apply the Mountain Pass Theorem to wave trains,
andpass to the limit L by means of concentrationcompactness. Moreover,they presenta different
→∞
existence proof for solitons which is based on the Nehari manifold of .
L
A new idea for convex potentials Φ was introduced by Filip and Venakides [FV99] by proposing to
maximise the potential energy under the convex constraint W with
γ
P ∈B
= W L2 : 1 W 2 γ .
Bγ { ∈ 2 k k2≤ }
A first advantage of this approachis that L< implies the functional to be continuous with respect
to the weak topology in L2, so that the existe∞nce of wave trains followPs from elementary principles of
infinite-dimensional convex analysis. Secondly, the improvement operator appears naturally in this
γ
T
context, and gives rise to effective approximation schemes for wave trains, see [Her05, DH08] and the
numerical simulations below.
Ourmethodisalsobasedontheconstrainedmaximizationofthepotentialenergybutyieldsimproved
results as it exploits several invariance properties .
γ
T
Constrained maximisation of the potential energy via the improvement operator
In view of (12) we formally define the improvement operator by
γ
T
√2γ
[W]:= ∂ [W],
γ
T ∂ [W] P
2
k P k
where the operator ∂ is the Gˆateaux derivative of , that means ∂ [W] = Φ( W) for all W L2.
′
P P P A A ∈
By construction, each fixed point W of is a travelling wave with 1 W 2= γ, and vice versa, where
Tγ 2 k k2
the speed is given by σ2 = ∂ [W] /√2γ.
2
k P k
Inthe nextSection 2.3we exploitthe convexityofΦ andderivethe followingbuilding blocksforthe
§
existence proof.
1. ∂ respects the positive cone
P
:= W L2 : W( ϕ )=W(ϕ )andW(ϕ ) W(ϕ )for almost all0 ϕ ϕ L ,
1 1 1 2 1 2
U { ∈ − ≥ ≤ ≤ ≤ }
which consists of all functions on [ L, L] that are even and unimodal. Moreover,for = ¯the
− − A A
operator ∂ also respects
P
:= W L2 : W(ϕ) 0for almost allϕ [ L, L] ,
N { ∈ ≥ ∈ − − }
which is the cone of all non-negative functions.
2. is well defined on and maps into ∂ , where
γ γ γ
T B \M B \M
:= W L2 : (W)=0
M ∈ P
is the set of all global minimisers of . (cid:8) (cid:9)
P
7
3. increases the potential energy, that means ( W) (W) for all W / , where equality
γ γ
T P T ≥ P ∈ M
holds if and only if W = [W].
γ
T
We are now able to describe the key principle that provides the existence of travelling waves.
Theorem 2.3. Let L2 be some positive cone that is invariant under the action of the operator ∂ .
S ⊂ P
Then, the set with = is invariant under the action of ,and each proper maximiser of
γ γ γ γ
S \M S S∩B T
in is a fixed point of , and hence a travelling wave.
γ γ
P S T
Proof. The invariance of is implied by the assumption on and the properties of . Now let W be
γ γ
S S T
a proper maximiser. Then, ( [W]) = (W) > min implies both [W] = W and W = , and
we conclude that W is in facPt aTγtravellingPwave with σP2 >|Sγ0. Tγ 6 M
In what follows the cone is given by either or . Since these cones are not open in L2 the
S U U ∩N
fact that each maximiser must satisfy the Euler-Lagrangeequation (12) with multiplier σ2 is not clear a
priori but provided by the invariance of under .
γ γ
S T
Theorem 2.3 yields only a sufficient condition for the existence of travelling waves. In fact, to show
that attains its maximum in is not trivial at all (at least in the soliton case), and requires a better
γ
P S
understanding of the energy landscape in . In our analysis we follow the direct approach and show
γ
that maximising sequences for are compSact in some appropriate topology in L2. More precisely, for
P
wavetrainsweuse weakcompactness,whereasinthe solitonsettingwe establishthe strongcompactness
for maximising sequences.
2.3 Some pieces of functional analysis
Here we prove the aforementioned properties of the improvement operator . To this end we rely on
γ
T
following standing assumptions on the potential Φ.
Assumption2.4. Forgivenγ >0weassumethattheinteractionpotentialΦhasthefollowing properties
on the interval [ √2γ, √2γ].
−
1. (Smoothness) Φ is at least C2,
2. (Convexity) Φ 0,
′′
≥
3. (Normalisation) 0=Φ(0)=Φ(0) and Φ (0)=β 0,
′ ′′
≥
4. (Non-triviality) Φ does not vanish identically.
The restriction to the interval [ √2γ, √2γ] is natural in our context, because W implies
γ
− ∈ B
W √2γ, see Lemma 2.5. As a consequence of Assumption 2.4 we find
kA k∞≤
0≤Φ(r)≤ 21r2(β+o(|r|)), Φ′(−|r|)≤0≤Φ′(|r|) (14)
for all r with r √2γ. Moreover, the non-triviality condition implies = W : (W) = 0 and
| | ≤ M { P }
= 0 , and hence each maximiser of in is proper.
γ γ
B \M6 { } P B
Within this section, the parameter L can take arbitrary values in (0, ], and Lp and W1,p with
∞
1 p denote the usual Lebesgue and Sobolev spaces on [ L, L], where
≤ ≤∞ −
L
W , W = W (ϕ)W (ϕ)dϕ
1 2 1 2
h i
Z
L
−
gives the inner product in L2.
8
Properties of the averaging operators ¯ and
A A
We summarize some elementary properties of the averagingoperators that are used in the proofs below.
b
Lemma 2.5. For any L, the operator ¯ is well defined on L2 and has the following properties:
A
1. ¯ maps into L2 L with ¯W W and ¯W W .
∞ 2 2 2
A ∩ kA k∞≤k k kA k ≤k k
2. ¯ maps into W1,2 with ¯W ′(ϕ)=W(ϕ+1/2) W(ϕ 1/2).
A A − −
3. ¯ is self-adjoint on L2. (cid:0) (cid:1)
A
4. If a sequence(W ) converges weakly in L2 to some limit W , then ¯W converges strongly in
n n ∞ A n n
L2([ L˜, L˜]) for each finite L˜ < with L˜ L. In particular, for L< the image of each bounded
set i−n L2 under the operator ¯ i∞s pre-com≤pact in L2 with respect to th(cid:0)∞e stro(cid:1)ng topology.
A
5. In the wave train case (L < ) the operator ¯ is compact. Moreover, the mth eigenvalue
∞ A
(m=0,1,2,...) is given by
mπ
̺ =Θ , Θ(̺):=̺ 1sin(̺),
m −
2L
(cid:16) (cid:17)
and the corresponding eigenspace is spanned by cos mπ and sin mπ
L · L ·
6. In the soliton case (L = ) the operator ¯ is no(cid:0)t com(cid:1)pact any(cid:0)more(cid:1), because it has continuous
spectrum specL2 ¯= Θ(̺)∞: ̺ R . In parAticular, specL2 ¯2 =[0, 1].
A { ∈ } A
Moreover, for L< we have :L2 L2 L with W W and W W .
∞ 2 2 2
∞ A → ∩ kA k∞≤k k kA k ≤k k
Proof. Definition (11) gives
b b b
L
¯V(ϕ)= χ¯(ϕ s)V(s)ds=χ¯(ϕ ) V (15)
A − −· ∗
Z
L
−
where χ¯ abbreviates the indicator function of the interval [ 1/2, 1/2], and denotes the convolution
− ∗
operator. H¨older’s inequality provides ¯V(ϕ) 2 ϕ+1/2V(s)2ds, and from this we readily derive the
A ≤ ϕ 1/2
firstassertion. Theproofsofthesecondandthethird−claimarethenstraightforward. Nowsupposethat
(cid:12) (cid:12) R
W W weaklyin L2([ L, L]), so th(cid:12)at ¯W(cid:12) ¯W point-wise thanks to (15), andthis implies the
n n
stron→g L2∞-convergence on−each finite intervAal [ L˜→, L˜A] du∞e to the uniform L -bounds.
∞
Towardsthespectralpropertiesof ¯westu−dyhowΘactsonplanewaves. Adirectcalculationshows
that each plane wave E (ϕ)=ϕ eiAkϕ satisfies the eigenvalue equation
k
7→
¯E =Θ(k/2)E
k k
A
pointwise, and this implies the fourth and the fifth assertion. Finally, for L< we have
∞
¯V(ϕ)=χ(ϕ ) V, χ(ϕ)=χ¯(ϕ) 1 .
A −· ∗ − 2L
This implies W(ϕ)2 χ χ¯(ϕ )W 2= ¯W(ϕ)2 and in turn the desired properties of .
|A | ≤k k∞k −b· k2 |A |b A
As a consequence of Definition (11) and Lemma 2.5 we easily find
b b b
1/2
kerL2 ¯= W L2 : W()=W( +1), W(ϕ)dϕ=0 . (16)
A ∈ · ·
n 1Z/2 o
−
In particular,the kernel of ¯is trivial if either L is irrational,or L= . Moreover,for L< we have
A ∞ ∞
kerL2 =kerL2 ¯ span 1 . (17)
A A⊕ { }
b
9
Properties of the potential energy functional
P
We rely on Assumption 2.4 and use standard methods from convex analysis to prove some properties of
and its derivative. All results are formulated in terms of the abstract averaging operator and hold
P A
both in the wave train and the soliton case.
Lemma 2.6. The functional is well-defined, bounded, continuous and Gaˆteaux-differentiable on ,
γ
and its derivative ∂ = ∂Φ P is a monotone operator and maps continuously into L2. MoreovBer,
γ
P A◦ ◦A B
for arbitrary W ,W we have
1 2 γ
∈B
(W ) (W ) m W W 2 + ∂ [W ], W W . (18)
P 2 −P 1 ≥ 2 kA 2−A 1k2 h P 1 2− 1i
and
∂ [W ] ∂ [W ], W W m W (ϕ) W (ϕ) 2 (19)
h P 2 − P 1 2− 1i≥ kA 2 −A 1 k2
where the monotonicity constant m is given by m=inf Φ (r) β.
|r|≤√2γ ′′ ≥
Proof. For all W in we have W √2γ, and Assumption 2.4 implies
γ
B kA k∞≤
Φ(r) C r , Φ(r) 1Cr2
| ′ |≤ | | ≤ 2
for all r with r √2γ, where C =sup Φ (r) β. Consequently, we find
| |≤ |r|≤√2γ| ′′ |≥
0 (W) 1C W 2 γC, ∂Φ[ W] Cp W Cp W ,
≤P ≤ 2 kA k2≤ kA A kp≤ k kp≤ k kp
and all assertions concerning the continuity and boundedness of both and ∂ follow immediately.
P P
Now let W , W be fixed, and notice that the convexity inequality (Φ(r ) Φ(r ))(r r )
1 2 γ ′ 2 ′ 1 2 1
∈ B − − ≥
m(r r )2 with r = W (ϕ) implies (19) by integration w.r.t. ϕ. To prove (18), let η [0, 1] and
2 1 i i
− ∈
consider W(η):=(1 η)W +ηW as well as
1 2 γ
− ∈B
p(η):= (W(η)).
P
The function p is well-defined and differentiable with respect to η, and using (19) we find
d
p(η)= ∂ [W(η)], W W
2 1
dη h P − i
η 1 ∂ [W(η)] ∂ [W ], ηW ηW + ∂ [W ], W W
− 1 2 1 1 2 1
≥ h P − P − i h P − i
=η−1 ∂ [W(η)] ∂ [W1], W(η) W1 + ∂ [W1], W2 W1
h P − P − i h P − i
≥η−1mkAW(η)−AW1k22 +h∂P[W1], W2−W1i
ηm W W 2 + ∂ [W ], W W .
≥ kA 2−A 1k2 h P 1 2− 1i
Finally, we integrate the last estimate from η =0 to η =1, and this gives (18).
The convexity of implies that each trivial travelling wave with σ2 = 0 must belong to , the set
P M
of all minimisers of .
P
Remark 2.7. We have ∂ [W]=0 for all W .
γ
P 6 ∈B \M
Proof. Assume there exists some W with ∂ [W] = 0. Then, (19) with W = 0 and W = W
γ 2 1
∈ B \M P
provides W =0 and hence (W)=0, which is a contradiction.
A P
Notice that for non-degenerate potentials Φ with Φ(r) > 0 for all r = 0 we have = ker , where
6 M A
ker is given in (16) and (17).
A
10