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ResearchCite this article: Xu H, Cuevas-Maraver J,Kevrekidis PG, Vainchtein A. 2018 Anenergy-based stability criterion for solitarytravelling waves in Hamiltonian lattices. Phil.Trans. R. Soc. A 376: 20170192.http://dx.doi.org/10.1098/rsta.2017.0192

Accepted: 7 November 2017

One contribution of 14 to a theme issue‘Stability of nonlinear waves and patterns andrelated topics’.

Subject Areas:applied mathematics, analysis

Keywords:solitary travelling wave, Hamiltonian lattice,stability

Author for correspondence:Haitao Xue-mail: [email protected]

Electronic supplementary material is availableonline at https://dx.doi.org/10.6084/m9.figshare.c.3992808.v1.

An energy-based stabilitycriterion for solitary travellingwaves in Hamiltonian latticesHaitao Xu1,2, Jesús Cuevas-Maraver3,4, Panayotis

G. Kevrekidis5 and Anna Vainchtein6

1Institute for Mathematics and its Applications, University ofMinnesota, Minneapolis, MN 55455, USA2Center for Mathematical Science, Huazhong University of Scienceand Technology, Wuhan, Hubei 430074, People’s Republic of China3Grupo de Física No Lineal, Departamento de Física Aplicada I,Universidad de Sevilla. Escuela Politécnica Superior, C/Virgen deÁfrica, 7, 41011-Sevilla, Spain4Instituto de Matemáticas de la Universidad de Sevilla (IMUS),Edificio Celestino Mutis. Avda. Reina Mercedes s/n, 41012-Sevilla,Spain5Department of Mathematics and Statistics, University ofMassachusetts, Amherst, MA 01003-9305, USA6Department of Mathematics, University of Pittsburgh, Pittsburgh,PA 15260, USA

HX, 0000-0003-0688-9966

In this work, we revisit a criterion, originallyproposed in Friesecke & Pego (Friesecke & Pego 2004Nonlinearity 17, 207–227. (doi:10.1088/0951715/17/1/013)), for the stability of solitary travelling waves inHamiltonian, infinite-dimensional lattice dynamicalsystems. We discuss the implications of this criterionfrom the point of view of stability theory, both atthe level of the spectral analysis of the advance-delay differential equations in the co-travellingframe, as well as at that of the Floquet problemarising when considering the travelling wave asa periodic orbit modulo shift. We establish thecorrespondence of these perspectives for the pertinenteigenvalue and Floquet multiplier and provideexplicit expressions for their dependence on thevelocity of the travelling wave in the vicinityof the critical point. Numerical results are usedto corroborate the relevant predictions in twodifferent models, where the stability may change

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rsta.royalsocietypublishing.orgPhil.Trans.R.Soc.A376:20170192

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twice. Some extensions, generalizations and future directions of this investigation are alsodiscussed.

This article is part of the theme issue ‘Stability of nonlinear waves and patterns and relatedtopics’.

1. IntroductionThe interplay of dispersion and nonlinearity in many spatially discrete physical and biologicalsystems often gives rise to solitary travelling waves [1–6]. In particular, such waves wereexperimentally observed in granular materials [7–10], and electrical transmission lines [3,11,12],among others. Following their discovery [13] in the Fermi–Pasta–Ulam (FPU) system [14–16],which revolutionized nonlinear science, there has been an enormous amount of literature onsolitary travelling waves in Hamiltonian lattices. Numerous significant developments includethe discovery of the integrable Toda lattice [1], existence results [17–19], fundamental studies ofthe low-energy near-sonic regime [20–25], where the waves are well described by the Korteweg–de Vries (KdV) equation, as well as the development of numerical [26–28] and quasi-continuum[29–33] approximations of such solutions.

An important issue that then naturally arises concerns establishing the necessary and sufficientconditions for the stability of solitary travelling waves. Stability results have been obtained insome special cases such as Toda lattice [34,35] or the near-sonic FPU problem [20–23], wherethe system’s integrability or near-integrability makes certain specialized techniques available.However, obtaining rigorous stability criteria in the general case of a non-integrable Hamiltonianlattice away from the KdV limit remains a challenge. While numerical simulations in manysystems suggest stability of solitary travelling waves, some recent studies [36–38] demonstratethe possibility of unstable waves when the energy (Hamiltonian) H of the wave decreases withits velocity c.

In a recent paper [39], inspired by and extending the fundamental work of [22], we examinedthe sufficient (but not necessary) condition for a change in the wave’s spectral stability occurringwhen the function H(c) changes its monotonicity. More specifically, we showed that when H′(c0) =0 for some critical velocity c0, a pair of eigenvalues associated with the travelling wave (in the co-travelling frame of reference in which the wave is steady) cross zero and emerge on the real axiswhen c is either above or below c0, thus creating instability. While this energy-based criterionfirst appeared in [22], where it was motivated by the study of the FPU problem in the near-soniclimit, in [39] we provided a concise proof as well as an explicit leading-order calculation for thesetwo near-zero eigenvalues. In addition, we tested the criterion numerically for a range of latticeproblems. This was accomplished by using two complementary approaches. The first approachinvolves the analysis of the spectrum of the linearized operator associated with travellingwaves as stationary solutions of the corresponding advance-delay partial differential equationin the co-travelling frame. The second approach is based on representing lattice travellingwaves as periodic orbits modulo shift and computing the corresponding Floquet multipliers.When the wave becomes unstable, a Floquet multiplier exits into the real line from the unitcircle. Importantly, through this computation, we identified the potentially unstable mode andcharacterized the growth rate of the instability when the mode becomes unstable. The examples in[39] illustrate that the results of the two numerical approaches match and agree with the analyticalpredictions. It should also be noted that the Floquet multiplier approach connects the establishedenergy-based criterion to the one recently obtained for discrete breathers [40].

In this paper, we present a significantly more detailed, systematic description of the analysisleading to the results in [39] and consider several non-trivial extensions, including an equivalentalgorithm for examining the energy criterion through the Floquet multiplier approach, anexamination of the higher-dimensional generalized kernel of the linearized operator and theconsideration of a non-generic but mathematically interesting case when the first and second



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derivatives of the energy function vanish simultaneously. Moreover, we discuss in detail howthe near-zero eigenvalues in the spectral problem depend to the leading order on the velocity ofthe solitary travelling wave near the critical point and investigate the corresponding conditions,through which we show the connection between the eigenvalues and the symmetries ofthe system.

We then illustrate these findings numerically by considering two examples, both of whichinvolve a non-monotone H(c), with H′(c)< 0 associated with unstable solitary travelling waves.One of the examples involves the α-FPU lattice with exponentially decaying long-range Kac–Baker interactions [37,41–43] and was also studied in [39] for different parameter values. Theother example concerns a smooth regularization of the FPU problem with piecewise quadraticpotential [38]. Unlike the original problem, for which our analysis framework is not available dueto the lack of regularity, the regularized model allows us to compute the eigenvalues and Floquetmultipliers. Both examples demonstrate the power of the energy-based criterion, with an excellentmatching of the results obtained by the two complementary approaches and a very good overallagreement with the theoretical predictions. In addition, the regularized model helps explain thenumerical stability results found earlier [38,39] for the problem with biquadratic potential.

The paper is organized as follows. In §2, we formulate the problem, introduce the functionspaces in which we will work and discuss some relevant properties of the Hamiltonian latticedynamical system. The energy-based criterion is derived in §3 by considering the eigenvalueproblem for the linearized operator, while the alternative perspective based on the Floquetanalysis is described in §4. Detailed asymptotic analysis of the eigenvalues splitting from zeronear the stability threshold is provided in §5. Section 6 illustrates and verifies the resultsnumerically via two examples. We discuss the results and identify future challenges in §7. Resultsof a more technical nature, numerical procedures used in the computations and discussion of theconnection to other energy-based criteria are included in the electronic supplementary material.

2. Problem formulation and preliminariesWe consider a one-dimensional Hamiltonian lattice with infinite-dimensional vectors q =(. . . , q−1, q0, q1, . . .)T and p = (. . . , p−1, p0, p1, . . .)T denoting its generalized coordinates andmomenta, respectively. The dynamics of the lattice is governed by

ddt

x(t) =A(x(t)) = J∂H∂x

, x(t) =(

q(t)p(t)

), J =

(0 I−I 0

), (2.1)

where H is the Hamiltonian energy density of the system I is the identity matrix and J is skew-symmetric. We assume that the lattice is homogeneous, i.e. (2.1) is invariant under integer shifts.To make it mathematically precise, we introduce the shift operators

S±yj = yj±1, S±y = (. . . , S±y−1, S±y0, S±y1, . . .)T, S±

(yz

)=(

S±yS±z

),

where y = (. . . , y−1, y0, y1, . . .)T, z = (. . . , z−1, z0, z1, . . .)T. We are interested in the systems thatsatisfy

A(S±x) = S±A(x).

If A is at least C1, differentiation yields that

DA(S±x) = S±DA(x)S∓, (2.2)

where D is the differential operator. If the Hamiltonian density depends on displacementsonly through strain variables r = (S+ − 1)q (rj = qj+1 − qj), the problem can be alternatively



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formulated asddt

y(t) = J1∂H∂y

, y(t) =(

r(t)p(t)

), J1 =

(0 S+ − I

I − S− 0

), (2.3)

where J1 is skew-symmetric. Below we focus on the formulation (2.1), but our arguments can alsobe applied to (2.3).

We now assume that system (2.1) has a family of solitary travelling wave solutions xtw(t; c)parametrized by non-zero velocity c. These are localized solutions of the form

xtw(t; c) =(

qtw(t; c)ptw(t; c)

), qtw,j(t; c) = q(j − ct), ptw,j(t; c) = p(j − ct), (2.4)

withlim

j→±∞qtw,j(t; c) = qtw,±∞(c), lim

j→±∞ptw,j(t; c) = 0 (2.5)

for any fixed t and c. In what follows, we assume for simplicity that qtw,±∞(c) = 0; note, however,that if either one of these values is non-zero (a kink-type solution), the strain formulation (2.3) canbe considered instead, with the (similarly defined) asymptotics rtw,±∞(c) = 0.

Note that the travelling wave solutions that we consider in (2.4) satisfy

xtw

(t + 1

c; c)

= S−xtw(t; c).

As a result, in order to fully define a travelling wave xtw,j(t; c) for all sites j ∈ Z and all time t ∈ R,it is sufficient to specify it for one site j = j0 and all time t ∈ R, or alternatively for all sites j ∈ Z andone period t ∈ [t0, t0 + 1/c).

In other words, a travelling wave solution is periodic modulo shift with period T = 1/c. With

this in mind, it is convenient to consider the map PT :(

qp

)�→(φψ

), where q and p are infinite-

dimensional functions defined on [0, T) while φ and ψ are functions from R to R, with φ(jT +t) = qj(t) and ψ(jT + t) = pj(t) for any j ∈ Z and t ∈ [0, T). In particular, we have P1/cxtw([0, 1/c)) =(

qtw,0(R)ptw,0(R)

)= xtw,0(R). Similarly, we define another map, PT :

(φ(t)ψ(t)

)�→(

q(t)p(t)

), where φ and ψ are

functions from R to R, while qj(t) = ejT∂tφ(t) = φ(jT + t) and pj(t) = ejT∂tψ(t) =ψ(jT + t) for j ∈ Z

and t ∈ [0, T). Here es∂τ is the translation operator such that (es∂τ Γ )(τ0) = Γ (τ0 + s). It is easy to seethat PT = (PT)−1 and P1/c(xtw,0(R)) = xtw([0, 1/c)).

Since our solutions are spatially localized, we can consider a finite-energy space such as

D1([0, T]) :=⎧⎨⎩(

q(t)p(t)

), t ∈ [0, T]

∣∣∣∣∣∣∫T

0

∑j∈Z

(|qj(t)|2 + |pj(t)|2 + |q′j(t)|2 + |p′

j(t)|2) dt<∞⎫⎬⎭ .

One can check that PT maps D1([0, T]) to

H1(R, R2) :={(

φ(t)ψ(t)

), t ∈ R

∣∣∣∣∫R

(|φ(t)|2 + |ψ(t)|2 + |φ′(t)|2 + |ψ ′(t)|2) dt<∞}

.

For the solutions that are exponentially localized in space, one can consider weighted spacessuch as

D1a ([0, T]) :=

⎧⎨⎩(

q(t)p(t)

), t ∈ [0, T]

∣∣∣∣∣∣∫T

0

∑j∈Z

(|qj(t)|2 + |pj(t)|2 + |q′j(t)|2 + |p′

j(t)|2) e2at dt<∞⎫⎬⎭ ,

which will be mapped to

H1a (R, R2) :=

{(φ(t)ψ(t)

), t ∈ R

∣∣∣∣∫R

(|φ(t)|2 + |ψ(t)|2 + |φ′(t)|2 + |ψ ′(t)|2) e2at dt<∞}

by PT.



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3. Energy-based criterion for spectral stability of solitary travelling wavesIn what follows we assume that xtw(t; c) is smooth in c and define

τ = ct, X(τ ) = x(t).

In the view of (2.1), X(τ ) satisfies

ddτ

X(τ ) = 1cA(X(τ )) = 1

cJ∂H∂X

. (3.1)

Note that, for given c,

Xtw(τ ; c) =(

Qtw(τ ; c)Ptw(τ ; c)

)= xtw(t; c)

and its derivatives ∂τ (Xtw(τ ; c)) and ∂c(Xtw(τ ; c)) are travelling waves with velocity 1, or,equivalently, these functions are periodic in τ modulo shift with period 1. Here we assumelimj→±∞ ∂k

c (Xtw,j(τ ; c)) = 0 for k = 1, 2, . . . , p (usually p = 2 or p = 3 is sufficient) such that∂k

c (Xtw,j(τ ; c)) for small k are still localized. Observe also that Θ =P1Xtw (where P1 =PT|T=1)satisfies

ddτΘ(τ ) = 1

c(P1)−1A(P−1

1 Θ) = 1cA(Θ). (3.2)

To explore spectral stability of a solitary travelling wave solution Xtw, we linearize (3.1) aroundit with X = Xtw + εY, obtaining

ddτ

Y(τ ) = 1c

DA(Xtw(τ ))Y(τ ) = 1c

J∂2H∂X2

∣∣∣∣∣X=Xtw

Y(τ ). (3.3)

We consider perturbations in the co-travelling frame of the form Y(τ ) = eλτZtw(τ ), where Ztw isa travelling wave with velocity 1, i.e. Ztw(τ ) = S+Ztw(τ + 1). This yields the eigenvalue problemfor the corresponding linear operator L:

λZtw =LZtw, L= 1c

DA(Xtw) − ddτ

= 1c

J∂2H∂X2

∣∣∣∣∣X=Xtw

− ddτ

. (3.4)

Note that existence of an eigenvalue with Re(λ)> 0 corresponds to instability. Equivalently, onecan linearize (3.2) around Θ using perturbation term eλτΓ (τ ) to obtain

λΓ = LΓ , L= 1c

DA(Θ) − ddτ

(3.5)

as an alternative representation of the eigenvalue problem.We now discuss the choice of the function space for the eigenvalue problem (3.4) (or,

equivalently, (3.5)). If we consider L as a closed operator in

D0([0, 1]) :=⎧⎨⎩x(t) =

(q(t)p(t)

), t ∈ [0, 1]

∣∣∣∣∣∣∫ 1

0

∑j∈Z

(|qj(t)|2 + |pj(t)|2) dt<∞⎫⎬⎭ ,

with its domain being D1([0, 1]) (which corresponds to L with domain in H1(R, R2)), the essentialspectrum of L usually contains the imaginary axis (e.g. the FPU case [22]) and the zero eigenvalue(whose existence is discussed below) is of course embedded in it. Since in this work we areinterested in the motion of the zero eigenvalue and its connection to the stability, we caneither obtain numerical insight on the (potential) effect of the essential spectrum from spectralcomputations in the case of a finite domain or attempt to separate the zero eigenvalue fromthe essential spectrum by working with weighted spaces, in order to carry out the relevantcalculations.

In what follows, we pursue the first approach. According to the numerical simulations forexamples presented in §6, the contribution from the essential spectrum to the motion of the zeroeigenvalue appears to be negligible at the leading order. Motivated by this a posteriori justification



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in what follows, we consider L in D1([0, 1]) and ignore the effects from the essential spectrum.While it is certainly worthwhile to analyse in detail these effects, which may result in higher-ordercontributions, such analysis is outside of the purview of the present study.

As an alternative strategy, one can move the essential spectrum away from the origin byconsidering weighted spaces such as D1

a([0, 1]). In this case, we can carry out the same calculationsand obtain the results presented below, provided two important modifications are made. Thefirst one concerns the inner product 〈·, ·〉 in D0([0, 1]). One can define the function B(Z, Z) =〈Z, Z〉D0([0,1]) that is continuous on D0

a([0, 1]) × D0−a([0, 1]) (since it is usually assumed that Z, Z ∈D0

|a|([0, 1]) = D0a([0, 1]) ∩ D0−a([0, 1])) and use this function B(·, ·) to replace the inner products in

D0([0, 1]). In particular, B(Z, J−1Z) = 0 corresponds to the conserved symplectic form [22]. As anexample showing how B will be used, the inner product 〈e1, J−1e0〉D0([0,1]) appearing in (3.9) and(3.10) below will become B(e1, J−1e0). Another issue is that for the same statements to hold in theframework of weighted spaces, our assumptions need to be suitably modified. For instance, ouranalysis below requires that the travelling wave solution Xtw, its derivatives ∂τXtw and ∂k

c Xtw

and generalized eigenfunctions of L for λ= 0 be in D0([0, 1]), but the use of weighted spacesrequires these functions to be in D0

a([0, 1]) (or D0|a|([0, 1])). This imposes a strong exponential

decay condition, which may not be satisfied by many of the existing solitary wave solutionsin lattices. For example, solutions in the example with long-range interactions presented in §6usually have algebraic decay. For this reason as well as those mentioned above, L with domainD1([0, 1]) is considered in the present work. Similar issues in a slightly different but related setting(a continuum KdV system near a point of soliton stability change) have been brought up in [44].

We now consider the properties of the linear operator L. Observe that L is a densely definedunbounded operator on the Hilbert space D0([0, 1]) with its domain being D1([0, 1]). Thus, theadjoint of L can be defined, and indeed we find that

L∗ = −J−1LJ, (3.6)

so that LJ and JL are self-adjoint (note that J−1 = −J). Observe also that the spectrum of L is2π i-periodic, because eλτZtw(τ ) = e(λ+i2nπ)τ (Ztw(τ ) e−i2nπτ ) and Ztw(τ ) e−i2nπτ is also a travellingwave with velocity 1. This symmetry is due to the fact that the shift operators S± commute withthe multiplier e2nπτ for n ∈ Z. Owing to the time-translation invariance of the solution Xtw, animportant property of L is that its kernel contains

e0 = ∂τXtw, (3.7)

as can be directly verified by differentiating (3.1) in τ and evaluating the result at X = Xtw.Here we assume the generic situation when ker(L) = span{e0}. Similarly, we have ker(L∗) =span{J−1e0}. Next, we observe that 〈e0, J−1e0〉 = 0, where 〈·, ·〉 denotes the inner product inD0([0, 1]). This implies that e0 ∈ (ker(L∗))⊥ = im (L), and thus there exists e1 ∈ D1([0, 1]) such thate0 =Le1, so that the algebraic multiplicity of the eigenvalue λ= 0 for L is at least two. In fact,differentiating (3.2) in c, we obtain cL(∂cXtw) = ∂τXtw, so

e1 = c∂cXtw (3.8)

satisfies Le1 = e0 and thus represents a generalized eigenvector of L associated with the zeroeigenvalue.

As we have seen, the spectrum of L always contains at least a double eigenvalue at zero.Following a similar argument, one finds that the algebraic multiplicity of the eigenvalue λ= 0 ishigher than two if and only if there exists e2 ∈ D1([0, 1]) such that Le2 = e1, which is equivalent to

〈e1, J−1e0〉 = 0. (3.9)

We will now prove that this can only happen when the derivative of the c-dependent conservedHamiltonian of the system,

H(c) =∫ 1

0H|Xtw(τ ;c) dτ ,



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is zero. Indeed, direct calculation shows that

0 = 〈e1, J−1e0〉 = 〈c∂cXtw, J−1∂τXtw〉 =⟨

c∂cXtw,1c∂H∂X

∣∣∣∣X=Xtw

⟩

=∫ 1

0(∂cXtw) ·

(∂H∂X

∣∣∣∣X=Xtw

)dτ =

∫ 1

0H′(c)|X=Xtw(τ ;c) dτ = H′(c). (3.10)

Assuming that H′(c) = 0, we note that as 〈e2, J−1e0〉 = 〈e2, J−1Le1〉 = 〈J−1Le2, e1〉 = 〈J−1e1, e1〉 = 0,e2 belongs to (ker(L∗))⊥ = im(L), and hence there exists e3 such that Le3 = e2. In other words,if the algebraic multiplicity of λ= 0 is higher than two, then it must be at least four. By similararguments, the algebraic multiplicity of the zero eigenvalue is always even, due to the symmetriesassociated with the Hamiltonian nature of the system.

Equation (3.10) implies that if H′(c) = 0 for some c = c0, the zero eigenvalue is at leastquadruple. Assume the generic case when H′′(c0) = 0. As soon as c deviates from the criticalvelocity c0, the above solvability condition (3.9) fails, and hence the second pair of eigenvalueswill start to move away from zero and generically emerge on the real axis when c is either below orabove the threshold value c0, indicating spectral instability of the corresponding travelling waves.This is confirmed by the relationship we derive in §5 between the sign of H′(c) and the leading-order behaviour of λ near zero (see theorem 5.3). Thus the condition H′(c) = 0 is a sufficientcondition for a change of the stability of a solitary travelling wave in Hamiltonian systems underconsideration. It should be noted that this condition is not necessary for the onset of instability.Indeed, a variety of different instability mechanisms may be available to the Hamiltonian system,including the Hamiltonian Hopf scenario, in which two eigenvalues may collide and give rise toa quartet associated with an oscillatory instability, or the period doubling case, where a pair ofmultipliers (see the Floquet analysis below) may exit the unit circle at −1. However, we will notsystematically explore these instability mechanisms in what follows.

We conclude this section with some technical results and notations that will be used below in§5. If H′(c) = 0, the 2-by-2 matrix M with entries given by Mj,k = 〈J−1ej, ek〉, j, k = 0, 1, is invertible.More precisely,

M =(

0 −α1α1 0

), α1 = 〈J−1e1, e0〉 = 0.

Let G2,0 = span{ei2nπτ ek | n ∈ Z, k = 0, 1}, G2 = cl(G2,0) and

G#2 = {X ∈ D0([0, 1]) | 〈X, J−1 ei2nπτ ek〉 = 0, n ∈ Z, k = 0, 1}.

Then D0([0, 1]) has the direct-sum decomposition D0([0, 1]) = G2 ⊕ G#2 [22].

When H′(c) = 0 and the zero eigenvalue is exactly quadruple, then 〈e0, J−1e1〉 = 〈e0, J−1e2〉 =〈e1, J−1e2〉 = 0 but 〈e0, J−1e3〉 = 0. Then

M = (〈J−1ej, ek〉)j,k=0,...,3 =

⎛⎜⎜⎜⎝

0 0 0 −α10 0 α1 00 −α1 0 −α2α1 0 α2 0

⎞⎟⎟⎟⎠

and α1 = 〈J−1e3, e0〉 = 〈J−1e1, e2〉 = 0, α2 = 〈J−1e3, e2〉.

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(3.11)

Observe that M is again invertible because α1 = 0 and we can explicitly find

M−1 = 1α1

⎛⎜⎜⎜⎜⎜⎝

0 −α2

α10 1

α2

α10 −1 0

0 1 0 0−1 0 0 0

⎞⎟⎟⎟⎟⎟⎠ .



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In this case, it is convenient to define G4,0 = span{ei2nπτ ek | n ∈ Z, k = 0, 1, 2, 3}, G4 = cl(G4,0) andG#

4 = {X ∈ D0([0, 1]) | 〈X, J−1 ei2nπτ ek〉 = 0, n ∈ Z, k = 0, 1, 2, 3}. Then D0([0, 1]) has the direct-sumdecomposition D0([0, 1]) = G4 ⊕ G#

4.In fact, one can easily extend the results above to the case where H′(c) = 0 and

the algebraic multiplicity of the zero eigenvalue is n, which, as we recall, must be aneven number. In this case, Lnen−1 =Ln−1en−2 =Ln−2en−3 = · · · =Le0 = 0, α1 = 〈J−1en−1, e0〉 =0 and M = (〈J−1ej, ek〉)j,k=0,...,n−1 is invertible. Similarly, we define Gn,0 = span{ei2nπτ ek | n ∈Z, k = 0, 1, 2, . . . , n − 1}, Gn = cl(Gn,0) and G#

n = {X ∈ D0([0, 1]) | 〈X, J−1 ei2nπτ ek〉 = 0, n ∈ Z, k =0, 1, 2, . . . , n}, and observe that D0([0, 1]) = Gn ⊕ G#

n.

4. Floquet analysis of the spectral stability of solitary travelling wavesIn this section, we consider an alternative approach for studying the spectral stability of solitarytravelling waves that is analytically equivalent to but numerically different from the eigenvalueproblem (3.4). It is based on the idea that the considered travelling waves on a lattice areperiodic orbits modulo shifts, and solitary travelling waves in particular can thus be identifiedwith localized time-periodic solutions known as discrete breathers. As a result, one expects anequivalence between the analysis of Floquet multipliers for the monodromy matrix associatedwith the relevant periodic orbit [45–47] and the eigenvalue problem (3.4), as discussed below.

We begin by defining the evolution operator E(τ0) such that E(τ0)X0 = X(τ0), where

cd

dτX(τ ) =A(X(τ )), τ ∈ [0, τ0]

and X(0) = X0.

⎫⎪⎬⎪⎭ (4.1)

Then travelling wave solutions of (3.1) satisfy E(1)Xtw(τ ) = S−Xtw(τ ) for any τ in [0, 1], and inparticular,

S+E(1)Xtw(0) = Xtw(0).

In other words, such solutions can be viewed as the fixed points of the map

F = S+E(1) − I. (4.2)

To study their stability, we consider another evolution operator E2(τ1, τ0) satisfying E2(τ1, τ0)Y0 =Y(τ1), where

cd

dτY(τ ) = DA(Xtw(τ ))Y(τ ), τ ∈ [τ0, τ1]

and Y(τ0) = Y0.

⎫⎪⎬⎪⎭ (4.3)

Let X0 = Xtw(0) + εY0, where εY0 is a small perturbation of the fixed point Xtw(0) of the map Fin (4.2). Substituting X0 into (4.1), we find that to the leading order

F (X0) = (S+E(1) − I)X0 = (S+E2(1, 0) − I)Y0.

Suppose now that Y0 satisfies E2(τ , 0)Y0 = eλτZtw(τ ), where Ztw is a travelling wave. Then onecan study the spectrum of S+E2(1, 0) to analyse the spectral stability of the solitary travellingwave solution Xtw. The proposition stated below shows that this spectrum is closely related tothe spectrum of the linear operator L defined in (3.4).

Proposition 4.1. Y0 is an eigenfunction of S+E2(1, 0) associated with the eigenvalue μ= eλ if and onlyif Z(τ ) = e−λτE2(τ , 0)Y0 is an eigenfunction of L associated with the eigenvalue λ.

The proof can be found in §I.A of the electronic supplementary material. Note that |μ|> 1implies instability and corresponds to the case Re (λ)> 0. The relationship between λ and μ

also explains why the spectrum of L is 2π i-periodic. We now state the result concerning thegeneralized eigenfunctions.



9


.........................................................

Proposition 4.2. If Z0(τ ) is an eigenfunction of L associated with the eigenvalue λ and (L − λI)kZk =(L − λI)k−1Zk−1 = · · · = (L − λI)Z1 = Z0, then Y0 = Z0(0) is an eigenfunction of S+E2(1, 0) associatedwith the eigenvalue μ= eλ, and there exist {Yj}1≤j≤k such that (S+E2(1, 0) − μI)kYk = (S+E2(1, 0) −μI)k−1Yk−1 = · · · = (S+E2(1, 0) − μI)Y1 = Y0. The converse is also true.

See §I.B of the electronic supplementary material for the proof. These results can be used tocompute {e0, e1, e2, e3} for L at c = c0 such that H′(c0) = 0 without solving the eigenvalue problem(3.4). To do so, we first find {Y0, Y1, Y2, Y3} such that (S+E2(1, 0) − I)Y0 = 0 and (S+E2(1, 0) −I)3Y3 = (S+E2(1, 0) − I)2Y2 = (S+E2(1, 0) − I)Y1 = Y0 and let Yj(0) =∑3

i=0Ω−1ji Yi for 0 ≤ j ≤ k, with

Ω satisfying

Ω

⎛⎜⎜⎜⎝

0 0 0 01 0 0 012 1 0 016

12 1 0

⎞⎟⎟⎟⎠=

⎛⎜⎜⎜⎝

0 0 0 01 0 0 00 1 0 00 0 1 0

⎞⎟⎟⎟⎠Ω .

For example, one can use

Ω =

⎛⎜⎜⎜⎝

1 0 0 00 1 0 00 1

2 1 0

0 13 −1 1

⎞⎟⎟⎟⎠ .

The readers can consult the proof of proposition 4.2 in §I.B of the electronic supplementarymaterial for the derivation of Ω and other details about this process. Let Yj(τ ) = E2(τ , 0)Yj(0) for0 ≤ j ≤ 3 and set

Z0(τ ) = Y0(τ ), Z1(τ ) = Y1(τ ) − τ Z0(τ ), Z2(τ ) = Y2(τ ) − τ Z1(τ ) − 12 τ

2Z0(τ )

and Z3(τ ) = Y3(τ ) − τ Z2(τ ) − 12 τ

2Z1(τ ) − 16 τ

3Z0(τ ).

We then seek ei in the form ei =∑3j=0 Fi,jZj, 0 ≤ i ≤ 3 and find F = (Fi,j)0≤i,j≤3 such that

e0 = ∂τXtw(τ ; c0), L3e3 =L2e2 =Le1 = e0.

In particular, let F0,0 = ∂τXtw(τ ; c0)/Z0(τ ); then Fi,j = δi,jF0,0 for 0 ≤ i, j ≤ 3.Suppose Ztw(τ ) is a travelling wave defined on R (although it is sufficient to define it on

[0, 1]) and we define E2(τ0) such that (E2(τ0)Ztw)(τ ) = E2(τ + τ0, τ )Ztw(τ ). It can be checked thatE2(τ0)Ztw is still a travelling wave. As e−s∂τ E2(τ + τ0 + s, τ + s) es∂τ = E2(τ + τ0, τ ), it follows thate−s∂τ E2(τ0 + s) es∂τ = E2(τ0). In fact, using arguments similar to the ones in [22], one can showe−s∂τ E2(s) is a group with the infinitesimal generator (1/c)DA(Xtw) − ∂τ =L. In other words,

e−s∂τ E2(s) = exp(

s(

1c

DA(Xtw) − ∂τ

))= exp(sL).

5. Splitting of the zero eigenvalueRecall that the spectrum of L has a double eigenvalue for any velocity c in the family of solitarytravelling wave solutions such that H′(c) = 0. We now consider the situation when there exists acritical non-zero velocity c0 such that H′(c0) = 0 but H′′(c0) = 0. As shown in §3, in this case themultiplicity of the zero eigenvalue of L at c = c0 is at least four, so that we generically expectthe onset of instability due to the additional two or more eigenvalues splitting away from zeroand appearing on the real axis. We start by considering the case when the eigenvalue is exactlyquadruple, i.e. there exist {e0, e1, e2, e3} such that L4e3 =L3e2 =L2e1 =Le0 = 0 and 〈 J−1e0, e3〉 = 0,and analyse the fate of the two eigenvalues splitting away from zero when c = 0.

Considering the neighbourhood of the critical velocity c = c0, we write c = c0 + ε and expand

Xtw(τ ; c) = U0 + εU1 + ε2U2 + ε3U3 + · · · , (5.1)



10


.........................................................

where we define U0 = Xtw(τ ; c0), U1 = (∂cXtw(τ ; c))|c=c0 , U2 = 12 (∂2

c Xtw(τ ; c))|c=c0 and U3 =16 (∂3

c Xtw(τ ; c))|c=c0 ; recall that we assumed that Xtw is smooth in c. Accordingly, the operator Lhas the expansion

L=L0 + εL1 + ε2L2 + ε3L3 . . . , Lk = 1k!

(∂

∂c

)kL|c=c0 , k = 0, 1, 2, . . . (5.2)

Recall that {e0, e1, e2, e3} are the eigenfunction and generalized eigenfunctions of L0 associatedwith λ= 0 and satisfying (3.11). In particular, e0 = ∂τU0 and e1 = c0U1. In what follows, we willuse G4 = cl(span{e0, e1, e2, e3}) and define the 4 × 4 matrices K, L, B and F with the entries

Kjk = 〈J−1ej,L1ek〉, Ljk = 〈J−1ej,L2ek〉,and Bjk = 〈J−1ej,L3ek〉, Fjk = 〈J−1ej,L4ek〉,

⎫⎬⎭ (5.3)

where j, k = 0, . . . , 3. Note that these matrices are symmetric because J−1 = −J = JT and JL is self-adjoint.

We now consider the behaviour of the two additional eigenvalues in the neighbourhood of c0.Below, we will show the following result.

Proposition 5.1. If the generalized kernel of L0 is exactly four-dimensional, the essential spectrum doesnot contain zero, and H′′(c0) = 0, then the splitting eigenvalues are λ=O(ε1/2).

We remark that this proposition holds in the framework with weighted spaces (in the settingwith unweighted spaces where zero is usually contained in the essential spectrum, the conditionsare usually not satisfied and one needs to investigate whether the effects of the essential spectrumcan be neglected). The result follows from the fact that two of the four eigenvalues of L are alwayszero. It suffices to calculate the leading-order terms of the eigenvalues for the perturbed operatorL at c = c0 + ε. By restricting the operator in the invariant subspace G4, the question reduces tothe perturbation of the matrix

A0 =

⎛⎜⎜⎜⎝

0 1 0 00 0 1 00 0 0 10 0 0 0

⎞⎟⎟⎟⎠ (5.4)

with two constraints that hold for any c,

L(∂τXtw(τ ; c)) = 0 (5.5)

andL(c∂cXtw(τ ; c)) = ∂τXtw(τ ; c). (5.6)

Note that the characteristic polynomial of the unperturbed matrix A0 is λ4 = 0. For the matrix A0with O(ε) perturbation, the characteristic polynomial is

λ4 + a3λ3 + a2λ

2 + a1λ+ a0 = 0, (5.7)

where the coefficients aj are at most O(ε). Moreover, due to two existing constraints (5.5) and (5.6),two of the eigenvalues are always zero, so we have λ2(λ2 + a3λ+ a2) = 0. Thus, either λ∼ ε1/2 (ifa2 = 0) or λ∼ ε (if a2 = 0). However, as we show below, a2 is proportional to H′′(c0) and thusnon-zero by assumption.

(a) Reduced eigenvalue problem without constraintsTo be more specific, let

Z|G4 = (e0, e1, e2, e3)

⎛⎜⎜⎜⎝

d0d1d2d3

⎞⎟⎟⎟⎠



11


.........................................................

and P =L|G4 where G4 is defined near the end of §3. Since D0([0, 1]) = G4 ⊕ G#4, here we consider

the decomposition of Z such that Z|G4 ∈ G4 and Z − Z|G4 ∈ G#4, while L|G4 is the operator L

restricted on the generalized kernel G4. Then the reduced eigenvalue problem is

P(e0, e1, e2, e3)

⎛⎜⎜⎜⎝

d0d1d2d3

⎞⎟⎟⎟⎠= λ(e0, e1, e2, e3)

⎛⎜⎜⎜⎝

d0d1d2d3

⎞⎟⎟⎟⎠ . (5.8)

Let A = A0 + εA1 + ε2A2 + ε3A3 . . . be the matrix representation of P in the basis G4, with theunperturbed matrix A0 given by (5.4). Then (5.8) can be written as

A

⎛⎜⎜⎜⎝

d0d1d2d3

⎞⎟⎟⎟⎠= λ

⎛⎜⎜⎜⎝

d0d1d2d3

⎞⎟⎟⎟⎠ . (5.9)

Projecting (5.8) onto J−1ej for j = 0, . . . , 3, we obtain

Q

⎛⎜⎜⎜⎝

d0d1d2d3

⎞⎟⎟⎟⎠= λM

⎛⎜⎜⎜⎝

d0d1d2d3

⎞⎟⎟⎟⎠ , (5.10)

where M is given by (3.11) and Q = (〈J−1ej, Pek〉) = (〈J−1ej,Lek〉), j, k = 0, . . . , 3, can be writtenas Q = Q0 + εQ1 + ε2Q2 + ε3Q3 + · · · . Note that Qi = MAi for i = 0, 1, . . . are symmetric, and inparticular, Q1 = K, Q2 = L and Q3 = B, where we recall (5.3). Considering the eigenvalue problem(5.9), we obtain the following result.

Proposition 5.2. Let Ai,jk denote the element in the j-th row, k-th column of Ai. For the eigenvalueproblem (5.9) without any constraints imposed,

— λ=O(ε1/2) if and only if

A1,30 = 0, (5.11)

A1,31 + A1,20 = 0 (5.12)

and (A1,32 + A1,21 + A1,10)2 + (A1,20A1,33 + A1,10A1,32 + A1,00A1,31 − A2,30)2 = 0; (5.13)

— λ=O(ε1/3) if and only if (5.11) holds but (5.12) fails;— λ=O(ε1/4) if and only if (5.11) fails.

Observe, however, that condition (5.12) always holds due to the symmetry of K, which impliesthat K01 = −α1A1,31 is equal to K10 = α1A1,20 (recall that α1 = 0). This eliminates the secondpossibility (λ=O(ε1/3)). As we will show below, the first constraint (5.5) ensures that (5.11) holds,thus eliminating the third possibility (λ=O(ε1/4)). Meanwhile, the second constraint (5.6) andour assumption that H′′(c0) = 0 together ensure that (5.13) holds, so we have λ=O(ε1/2).

(b) Reduced eigenvalue problem with constraints and the leading approximationfor the splitting eigenvalues

In this section, we use the constraints (5.5) and (5.6) to show that the splitting eigenvalues are ofO(ε1/2) and compute their leading approximation.

Since (5.5) holds for any c, expansion around c = c0 using (5.1) and (5.2) shows that

0 =L0(∂τU0),

0 =L0(∂τU1) + L1(∂τU0),

0 =L0(∂τU2) + L1(∂τU1) + L2(∂τU0)

and 0 =L0(∂τU3) + L1(∂τU2) + L2(∂τU1) + L3(∂τU0).



12


.........................................................

Using

∂τUj =3∑

k=0

(gjkek) + (∂τUj)#, (∂τUj)

# ∈ G⊥4 , (5.14)

we can rewrite the above equations as

0 = Q0g0, (5.15)

0 = Q0g1 + Q1g0, (5.16)

0 = Q0g2 + Q1g1 + Q2g0 (5.17)

and 0 = Q0g3 + Q1g2 + Q2g1 + Q3g0. (5.18)

Since e0 = ∂τU0, we have g0 = (g00, g01, g02, g03)T = (1, 0, 0, 0)T. Recalling that Q0 = MA0, Q1 = K,Q2 = L and Q3 = B, we then use (5.16) to find that

K00 = 0, (5.19)

which corresponds to (5.11), and

g1 = −AT0 M−1Q1g0 + g10g0, (5.20)

which yields

g11 = −K30

α1+ α2

α21

K10, g12 = K20

α1and g13 = −K10

α1. (5.21)

Using (5.17), we then obtain3∑

j=1

g1jK0j + L00 = 0, (5.22)

which corresponds to

A1,20A1,33 + A1,10A1,32 + A1,00A1,31 − A2,30 = 0, (5.23)

and the coefficients g2 = −AT0 M−1Q1g1 − AT

0 M−1Q2g0 + g20g0, given in detail by (21) in §II of theelectronic supplementary material.

Finally, (5.18) yields3∑

j=1

g2jK0j +3∑

j=0

g1jL0j + B00 = 0 (5.24)

and the formulae (22) in the electronic supplementary material for the coefficients g3 =−AT

0 M−1Q1g2 − AT0 M−1Q2g1 − AT

0 M−1Q3g0 + g30g0.We now turn to the constraint (5.6), which again holds for any c. Expanding both sides in ε and

using (5.1) and (5.2), we obtain

L0(c0U1) = ∂τU0,

L0(2c0U2 + U1) + L1(c0U1) = ∂τU1,

L0(3c0U3 + 2U2) + L1(2c0U2 + U1) + L2(c0U1) = ∂τU2

and L0(4c0U4 + 3U3) + L1(3c0U3 + 2U2) + L2(2c0U2 + U1) + L3(c0U1) = ∂τU3.

Using

Uj =3∑

k=0

( fjkek) + (Uj)#, (Uj)

# ∈ G⊥4 , (5.25)



13


.........................................................

we have

c0Q0f1 = Mg0, (5.26)

Q0(2c0f2 + f1) + Q1c0f1 = Mg1, (5.27)

Q0(3c0f3 + 2f2) + Q1(2c0f2 + f1) + Q2c0f1 = Mg2 (5.28)

and Q0(4c0f4 + 3f3) + Q1(3c0f3 + 2f1) + Q2(2c0f2 + f1) + Q3c0f1 = Mg3. (5.29)

As shown in §II of the electronic supplementary material, this yields the coefficients fij, i, j =1, . . . , 3 in (5.25). In particular, we have

f23 = K20 − K11 − c0f10K10

2α1c0. (5.30)

Recall that proposition 5.2 states that λ=O(ε1/2) if and only if (5.11)–(5.13) hold. The aboveanalysis shows that (5.11) is imposed by the constraint (5.5). As we discussed above, (5.12) alsoholds, due to the symmetry of K. Observe also that (5.23) derived above means that (5.13) holdsif and only if

A1,32 + A1,21 + A1,10 = 0. (5.31)

We now show that this inequality holds due to our assumption that H′′(c0) = 0, so that byproposition 5.2 we have λ=O(ε1/2). Indeed, writing H(c) = H(c0) + εH′(c0) + (ε2/2)H′′(c0) +(ε3/6)H′′′(c0) + o(ε3) and recalling (3.1), (3.11), (5.1), (5.14), (5.21), (5.25) and (5.30), we obtain

H′′(c0) = 2〈∇H, U2〉 + 〈U1, ∇2HU1〉= 2c0〈J−1e0, U2〉 + c0〈U1, J−1L0U1〉 + c0〈U1, J−1∂τU1〉= 2c0f23〈J−1e0, e3〉 + 〈U1, J−1e0〉 + c0f10g13〈e0, J−1e3〉 + g12〈e1, J−1e2〉

= 2c0K20 − K11 − c0f10K10

2α1c0(−α1) + 0 − c0f10K10 − K20

= K11 − 2K20 = α1(A1,21 + A1,32 + A1,10), (5.32)

where

∇H= ∂H∂X

∣∣∣∣X=Xtw(τ ;c0)

and ∇2H= ∂2H∂X2

∣∣∣∣∣X=Xtw(τ ,c0)

. (5.33)

Here we used the fact that K11 = α1A1,21 and K02 = −α1A1,32 = −α1A1,10 = K20, where the lastequality takes into account (5.11). Recall also that α1 defined in (3.11) is non-zero.

Substituting λ= ε1/2λ1 + ελ2 + ε3/2λ3 + · · · and Z = Z0 + ε1/2Z1 + εZ2 + ε3/2Z3 + · · · intoλZ =LZ, we obtain

0 =L0Z0,

λ1Z0 =L0Z1,

λ1Z1 + λ2Z0 =L0Z2 + L1Z0,

λ1Z2 + λ2Z1 + λ3Z0 =L0Z3 + L1Z1

and λ1Z3 + λ2Z2 + λ3Z1 + λ4Z0 =L0Z4 + L1Z2 + L2Z0.

With Zj =∑3k=0(Cjkek) + (Zj)#, where (Zj)# ∈ G⊥

4 , we then have

0 = A0C0, (5.34)

λ1C0 = A0C1, (5.35)

λ1C1 + λ2C0 = A0C2 + M−1Q1C0, (5.36)

λ1C2 + λ2C1 + λ3C0 = A0C3 + M−1Q1C1 (5.37)

and λ1C3 + λ2C2 + λ3C1 + λ4C0 = A0C4 + M−1Q1C2 + M−1Q2C0, (5.38)



14


.........................................................

where we recall that M defined in (3.11) is skew-symmetric, while Q0 = MA0, Q1 = K, Q2 = L andQ3 = B are symmetric. Setting C00 = 1, we find that

C0 = C00g0 = g0,

C1 = λ1AT0 C0 + C10g0 = λ1AT

0 g0 + C10g0,

C2 = AT0 (λ1C1 + λ2C0 − M−1KC0) + C20g0

and C3 = AT0 (λ1C2 + λ2C1 + λ3C0 − M−1KC1) + C30g0.

Owing to the special form of A0, we first check that the last row of each system in (5.34)–(5.37)is satisfied. The first two are straightforward. Observe that the last row of (5.36) is equivalent tothe last row of λ1C1 + λ2C0 = M−1KC0, or 0 = M−1KC0, because the last components in C0 andC1 are zero, and yields (5.19). In addition, the last row of (5.37) is equivalent to the last row ofλ1C2 = M−1KC1, which can be checked using (5.27). The last row of (5.38) then yields

α1λ41 + (2K20 − K11)λ2

1 +(

L00 + K220α1

+ α2K210

α21

− 2K10K30

α1

)= 0,

where the last term is zero by (5.22), as can be seen after substituting (5.21) and taking into accountthe symmetry of K, and the coefficient in front of λ2

1 in the second term is −H′′(c0) by (5.32) (forfurther discussion about the connection between the last row of the equations and the constraints(5.5) and (5.6), see §II of the electronic supplementary material).

We thus obtain the particularly simple final form

λ21(α1λ

21 − H′′(c0)) = 0,

which has a double root at zero, corresponding to two eigenvalues that are always zero, and anon-zero pair of roots satisfying λ2

1 = H′′(c0)/α1. Clearly, H′′(c0)α1 < 0 thus corresponds to a pairof imaginary λ1 and H′′(c0)α1 > 0 yields a pair of real ones, implying the transition to instabilityas c passes through c0. To be more specific, if H′(c0) = 0 and H′′(c0)α1 > 0, as c increases fromc1 < c0 to c0, there will be a pair of eigenvalues λ≈ ±i

√(H′′(c0)/α1)(c0 − c) moving towards each

other along the imaginary axis and colliding at the origin at c = c0. As we continue increasing c toc2 > c0, a pair of zero eigenvalues will split and move on the real axis as λ≈ ±√(H′′(c0)/α1)(c − c0).If H′′(c0)α1 < 0, this process will be reversed as c increases.

We summarize these results in the following theorem.

Theorem 5.3. Consider a lattice Hamiltonian system dx/dt = J(∂H/∂x) and assume that

— it has a family of travelling wave solutions xtw(t; c) parametrized by velocity c that are smoothin c;

— for τ = ct and Xtw(τ ; c) = xtw(t; c), we have ∂τ (Xtw,j(τ ; c)) and ∂kc (Xtw,j(τ ; c)) in D0([0, 1]) for

k = 1, 2 (if weighted spaces are used, these functions should be in D0a([0, 1]));

— there exists velocity c0 such that the system’s Hamiltonian H(c) = ∫10 H|X=Xtw(τ ;c) dτ satisfies

H′(c0) = 0 and H′′(c0) = 0;— the generalized kernel of L0 = (1/c)J(∂2H/∂X2)|X=Xtw(τ ;c0) is of dimension 4, with L4

0e3 =L3

0e2 =L20e1 =L0e0 = 0;

— the contribution from the essential spectrum to the zero eigenvalue is at a higher order (thisassumption is not needed if weighted spaces are used).

Then there exist c1 < c0 and c2 > c0 such that, for c ∈ (c1, c2), a pair of eigenvalues of L is given by

λ= ±√

H′′(c0)α1

(c − c0) + O(|c − c0|),

where α1 = 〈J−1e3, e0〉. In particular, if H′′(c0)α1 < 0, the solution xtw(t; c) is unstable for c ∈ (c1, c0).If H′′(c0)α1 > 0, the solution xtw(t; c) is unstable for c ∈ (c0, c2).



15


.........................................................

(c) Higher-dimensional generalized kernelWe now generalize theorem 5.3 to the case when the generalized kernel of L0 has dimensionlarger than or equal to 4: gker(L0) = n, n ≥ 4. Recall that n must be even.

Lemma 5.4. Suppose all the assumptions in theorem 5.3 hold except that dim(gker(L0)) = n ≥ 4, wheren is even, and

Ln0en−1 =Ln−1

0 en−2 = · · · =L20e1 =L0e0 = 0.

Then there exist c1 < c0 and c2 > c0 such that, for c ∈ (c1, c2), the leading-order terms of non-zeroeigenvalues of L are

λ= ej(2π i/(n−2))(

H′′(c0)α1

(c − c0))1/(n−2)

+ o(|c − c0|1/(n−2)), j = 0, 1, . . . , n − 3,

where α1 = 〈J−1en−1, e0〉.

The proof can be found in §I.C of the electronic supplementary material.The degenerate case when H′(c0) = H′′(c0) = 0 but H′′′(c0) = 0 is briefly discussed in §III of the

electronic supplementary material.

6. ExamplesIn this section, we illustrate the results of our stability analysis by considering two examples oflattice dynamical systems that have solitary wave solutions which change stability. Details of thenumerical procedures used to obtain these solutions and analyse their stability can be found in§IV of the electronic supplementary material.

As our general set-up, we consider a lattice of coupled nonlinear oscillators with a genericnearest-neighbour potential and all-to-all harmonic long-range interactions. In principle, themethodology can capture nonlinear long-range interactions, but here we consider linear onesfor simplicity. Such a system is described by the following Hamiltonian:

H =∞∑

n=−∞

[12

q2n + V(qn+1 − qn) + 1

4

∞∑m=−∞

Λ(m)(qn − qn+m)2

], (6.1)

where qn(t) represents the displacement of the nth particle from the equilibrium position at timet, qn(t) ≡ q′

n(t), V(r) is a generic potential governing the nonlinear interactions between nearestneighbours, and Λ(m) are long-range interaction coefficients, which decay as |m| → ∞; in theabsence of such interactions, Λ(m) = 0. For instance, Λ(m) = ρ(eγ − 1) e−γ |m|(1 − δm,0), where ρ >0 and γ > 0 are constants, corresponds to the Kac–Baker interaction [37], and Λ(m) = ρ|m|−s(1 −δm,0) corresponds to the power law interaction between charged particles on a lattice, with s = 5and s = 3 as special cases of interest [36]. The dynamics of the system is governed by

qn − V′(qn+1 − qn) + V′(qn − qn−1) +∞∑

m=1

Λ(m)(2qn − qn+m − qn−m) = 0. (6.2)

Since the solitary solutions we consider are kink-like in terms of displacement, it is moreconvenient to rewrite (6.2) in terms of the strain variables rn = qn+1 − qn, obtaining

rn + 2V′(rn) − V′(rn+1) − V′(rn−1) +∞∑

m=1

Λ(m)(2rn − rn+m − rn−m) = 0. (6.3)

Solitary travelling wave solutions of (6.3) have the form

rn(t) = φ(ξ ), ξ = n − ct,



16


.........................................................

where c is the velocity of the wave. Such solutions thus satisfy the advance-delay differentialequation

c2φ′′(ξ ) + 2V′(φ(ξ )) − V′(φ(ξ + 1)) − V′(φ(ξ − 1)) +∞∑

m=1

Λ(m)(2φ(ξ ) − φ(ξ + m) − φ(ξ − m)) = 0.

(6.4)

In addition, the solutions must vanish at infinity:

limξ→±∞

φ(ξ ) = 0. (6.5)

As our first example of a system where stability of a solitary travelling wave changes withvelocity, we consider a smooth regularization of the model studied in [38]. The original model isan FPU lattice (6.1) with only short-range interactions (Λ(m) = 0) and a biquadratic potential,

V(r) =

⎧⎪⎪⎨⎪⎪⎩

r2

2, |r| ≤ rc

χ

2(|r| − rc)2 + rc|r| − r2

c2

, |r|> rc,

(6.6)

which allows construction of explicit solitary travelling waves [38]. Here rc > 0 is a fixed transitionstrain separating the quadratic regimes, and χ > 1 is the elastic modulus for |r|> rc, while theother modulus is rescaled to one. As shown in [38], the system has a family of solitary travellingwave solutions rn(t) = φ(n − ct) with velocities c in the range 1< c<

√χ . As c approaches the

lower sonic limit, c → 1, solutions delocalize, with rn(t) → rc > 0 for all n. As a result, essentiallyby construction, the energy H(c) of the system tends to infinity in this limit. This is in contrast tothe smoother FPU systems, where solutions delocalize to zero as they approach the sonic limit,with H(c) → 0 and are well described by the KdV equation [20]. In this case, however, the systemis linear for |r|< rc and thus cannot have nonlinear waves delocalizing at zero. As the velocity cincreases away from c = 1, the wave becomes more localized, and the energy H(c) decreases for cjust above the sonic limit. However, its amplitude also increases, so eventually the energy startsgrowing with c and tends to infinity as c → √

χ . As a result, H(c) has a minimum at velocity c = c0,with H′(c0) = 0, H′(c)< 0 for 1< c< c0 and H′(c)> 0 for c0 < c<

√χ . According to our energy-

based criterion, we thus expect to see a stability change at c = c0. This is confirmed by directnumerical (time evolution) simulations in [38,39], which indicate that the solitary travelling wavesare unstable when 1< c< c0 and stable for c0 < c<

√χ .

The potential (6.6) has a discontinuous second derivative:

V′′(r) ={

1, |r| ≤ rc

χ , |r|> rc,(6.7)

so our analysis framework is not directly applicable in this case. Instead, we consider a smoothregularization of this potential, with V′′(r) in the form

V′′(r) = 1 + χ − 1π

(arctan

r2 − r2c

ε2 + arctanr2

c

ε2

), (6.8)

where ε > 0 is the regularization parameter. The expression for V(r) is quite cumbersome and thusnot included here. Figure 1 shows the shape of V′(r) and V′′(r) for different values of ε, includingε= 0 when V(r) reduces to (6.6). Note that, for all ε≥ 0, we have the same lower sonic limit:cs = V′′(0) = 1. Increasing ε from zero smoothens the corners of V′(r) ar r = ±rc and decreases itsslope at |r|> rc.

We used spectral methods (see §IV of the electronic supplementary material for details) toconstruct a family of solitary travelling waves for the regularized system, with typical profilesshown for the strain variable in figure 2a. Importantly, the smooth regularized potential makespossible the regular near-sonic KdV limit, with solutions delocalizing to zero and H(c) → 0



17


.........................................................

0.5 1.0 1.5 2.00

1

2

3

4

5

r0 0.5 1.0 1.5 2.0

r

V¢ (

r)

biquadratice = 0.1e = 0.25

1.0

1.5

2.0

2.5

3.0

3.5

4.0

V≤

(r)

(b)(a)

Figure 1. First (a) and second (b) derivatives of the regularized potential V(r) for different values of the parameter ε. Hererc = 1 andχ = 4, and only r ≥ 0 is shown due to symmetry. (Online version in colour.)

–20 –10 0 10 200

0.5

1.0

1.5

x

f (x

)

c = 1.005c = 1.03c = 1.08

–100 –50 0 50 100–2.5

–2.0

–1.5

–1.0

–0.5

0

x

c = 1.65c = 1.7c = 1.8

(b)(a)

Figure 2. Typical profilesφ(ξ ) of solitary travelling waves for the regularized potential with ε= 0.25,χ = 4 and rc = 1 (a)and theα-FPU model with Kac–Baker long-range interactions, for γ = 0.18 andρ = 0.02 (b). (Online version in colour.)

as c → 1. For small enough ε, we have a narrow interval 1< c< c0,max, where H(c) rapidlyincreases from zero. The energy then reaches a local maximum value at c = c0,max (H′(c0,max) =0, H′′(c0,max)< 0) and decreases for c0,max < c< c0,min to a local minimum value at c = c0,min(H′(c0,min) = 0, H′′(c0,min)> 0) and then starts increasing again. As ε tends to zero, we havec0,max → 1, c0,min → c0, and the local maximum value H(c0,max) tends to infinity. This is illustratedin figure 3a.

The regularization also allows to analyse the spectral stability of the solitary travelling wavesolutions using the two approaches described in §§3–5, co-travelling steady-state eigenvalueanalysis and calculation of Floquet multipliers for periodic orbits (modulo shift). Figure 3showcases the power of the stability criterion and illustrates the complementary nature of thetwo approaches. For the parameter values ε= 0.25, χ = 4 and rc = 1, we have c0,max = 1.00546 andc0,min = 1.0497 such that H′(c)< 0 for c0,max < c< c0,min, and H′(c)> 0 otherwise. In the velocityinterval (c0,max, c0,min), an eigenvalue of the operator L crosses through λ= 0 into the positive realaxis (dots in figure 3b), indicating instability. In fact, it can be shown [22] that the stability problemin the co-travelling frame also possesses eigenvalues λ+ i(2π j), where j ∈ Z, as demonstrated infigure 4a. Note that the finite domain nature of the computation leads to a number of spurious



18


.........................................................

1.00 1.01 1.02 1.03 1.04 1.05 1.060

1

2

3

4

5

6

7

8

9

c

H (

c)

biquadratice = 0.1e = 0.25

1.03 1.035 1.04 1.045 1.05 1.055 1.06 1.065 1.072.30

2.32

2.34

2.36

2.38

2.40

1.00 1.02 1.04 1.06 1.080

0.01

0.02

0.03

0.04

0.05

0.06

l

c

(b)(a)

Figure 3. Travelling solitary waves for the regularized potential with χ = 4, rc = 1. (a) The energy-velocity dependenceH(c), with H′(c)> 0 implying (spectral) stability, and H′(c)< 0 implying instability. Panel (b) confirms this by showing themaximum real eigenvalue obtained by diagonalizing the linearization operatorL (dots) and transforming the relevant Floquetmultiplierμ into a corresponding eigenvalue (for comparison) via the relationλ= log(μ) (solid curve) for ε= 0.35. Verticallines in (b) indicate the values of c where monotonicity of H(c) changes. (Online version in colour.)

–0.03 –0.02 –0.01 0 0.01 0.02 0.03–3

–2

–1

0

1

2

3

Im(l

)

Re (l)

–0.04 –0.02 0 0.02 0.04–150

–100

–50

0

50

100

150

–1.0 –0.5 0 0.5 1.0

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

Im(m

)

Re (m)

(b)(a)

Figure 4. Stability analysis of the unstable solitary travelling wave with velocity c = 1.034 for the regularized potential withε= 0.25, χ = 4, rc = 1. (a) The spectrum of the linearization operatorL for Im (λ) ∈ (−π ,π ); the inset corresponds tothe full spectrum. In (b), the Floquet multipliers spectrum is shown. (Online version in colour.)

unstable eigenvalues in the figure. However, we have systematically checked (see also §IV ofthe electronic supplementary material for relevant details) that their real part decreases with thesystem length and hence expect such spurious instabilities to be absent in the infinite domainlimit. Finally, the solid curve in figure 3b shows the Floquet multiplier calculation associatedwith the time T = 1/c map of the corresponding periodic orbit, transformed (in order to comparewith the steady-state eigenvalue approach) according to the relation λ= log(μ). Confirming thecomplementary picture put forth, we find that, in this case, a pair of Floquet multipliers crossesthrough (1, 0) and into the real axis for the exact same parametric interval. Figure 4b depicts thetypical Floquet spectrum for an unstable travelling wave. Note that there is a slight mismatchbetween the bifurcation loci predicted by the local extrema of H(c) (denoted by c0,max and c0,min)and the values predicted by the eigenvalues of the linearization operator L (denoted by cl

0,max

and cl0,min, respectively). When the system length l is increased, c0,max and c0,min remain invariant



19


.........................................................

1.00569 1.00570 1.00571 1.00572 1.00573 1.00574 1.00575–10

–8

–6

–4

–2

0

2

4

6

8l2

(×10

–9)

c

1.0489 1.0490 1.0491 1.0492 1.0493 1.0494–1.0

–0.5

0

0.5

1.0

1.5

l2 (×

10–6

)

c

(b)(a)

Figure 5. Dependence of the squared linearization eigenvalueλ2 on c for solitary wave solutions for the regularized potentialwith ε= 0.25,χ = 4, rc = 1 in the vicinity of bifurcations at c0,max (a) and c0,min (b). The straight line in each panel representsthe best linear fit from which the corresponding value ofβ is inferred numerically. (Online version in colour.)

Table 1. Comparison between the numerical slopeβ and its theoretical predictionβt in (6.9) of theλ2 versus c dependence atthe bifurcations c0,max and c0,min in the problemwith the regularized potential (ε= 0.25,χ = 4, rc = 1). Parameters involvedin the theoretical calculations are also shown. Here the first column shows the values of c0 at which H′(c0)= 0 (c0,max in thefirst row and c0,min in the second), and the second column shows the corresponding values of cl0 where an eigenvalue crosseszero (or a Floquet multiplier crosses 1), together with H′′(c0) andα1.

c0 cl0 H′′(c0) α1 βt β

1.00546 1.00572 −5.3912 × 104 −5.6315 × 107 9.5734 × 10−4 2.4400 × 10−4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.0497 1.0492 3.2588 × 102 −7.0461 × 104 −4.6250 × 10−3 −4.5713 × 10−3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

but cl0 approaches the corresponding c0 values. This is a consequence of the fact that, for small l,

the solution at the boundaries is larger than the machine precision.To connect these results with the theoretical analysis of (3.4), figure 5 shows the dependence of

λ2 with respect to c near cl0 for both bifurcations, which according to theorem 5.3, must be linear

in the vicinity of cl0 with the slope

βt = H′′(c0)α1

, (6.9)

where α1 is defined in (3.11). To compute βt in (6.9), we use c0, i.e. the value of c at which H′(c) = 0;however, for determining α1 we calculate the eigenmodes at cl

0, as this value corresponds to thepoint where we are closer to a quadruplet of zero eigenvalues. Table 1 shows the numericalvalue of the slope β (obtained by the best linear fit) and its theoretical prediction (6.9) for bothbifurcations, at c0,max and c0,min, where α1 is computed numerically. There is a mismatch ofapproximately 1% at the second bifurcation, at c0,min. This mismatch is probably due to the factthat α1 in theorem 5.3 cannot be computed at the precise value of cl

0 in the numerical set-up.The theoretical prediction βt at the first bifurcation at c0,max is less accurate in comparison tothe numerical value of β (although it has the same order of magnitude). This is likely caused bythe fact that the solitary travelling wave is rather wide at the bifurcation (recall that the solitarywaves delocalize when c approaches the sonic limit cs = 1, as illustrated in figure 2), and Le1and L2e2 are not precisely in the kernel of L, causing an error in computing α1 and thereforeβt. We note that, although the value l = 100 has been fixed for the length of the system in thecalculations leading to figure 4 and table 1, we used different values of the mesh size �ξ (see§IV of the electronic supplementary material for details) at the two bifurcations,�ξ = 0.025 at the



20


.........................................................

t/T

n100 200 300 400 500 6000

100

200

300

400

500

0 100 200 300 400 500 600–0.4

–0.2

0

0.2

0.4

0.6

r n (t

= 5

00T

)

n

(b)(a)

Figure 6. Evolution of the unstable solitary travelling wave with c = 1.034 for the regularized potential whose spectrum isrepresented in figure 4. (a) The space–time evolution dynamics of the strains; (b) the strain profile at t = 500T , where T = 1/c.Here ε= 0.25,χ = 4 and rc = 1. The decay of the structure through the creation of a state reminiscent of a dispersive shockwave [48] is evident. (Online version in colour.)

first one and �ξ = 0.0125 at the second. These choices were motivated by the fact that at highc the solitary wave must be resolved with a finer grid in order to feel the regularization of thepotential. Better results for the theoretical predictions at the first bifurcation could be obtainedby substantially increasing the system’s length; however, this would render the linearizationproblem rather unwieldy.

Despite the numerical issues discussed above, these results are generally in very goodagreement with the theoretical predictions. In addition to showing instability when H′(c)< 0 andthe linear dependence of the squared eigenvalue on c − c0, in agreement with theorem 5.3, theyalso suggest that H′(c)> 0 corresponds to spectral stability. In particular, the solitary travellingwaves are apparently stable at 1< c< c0,max, which is consistent with stability of near-sonic wavesproved in [23]. This example also provides a nice explanation of the instability of solitary wavesnear the lower sonic limit c = 1 at ε= 0, when c0,max = 1, and the problem no longer has a KdVlimit (unless one allows rc and χ to depend on c [38]).

To complement these results by means of direct numerical simulations, we now considerthe dynamic evolution of unstable travelling waves by solving (6.3) numerically (see §IV ofthe electronic supplementary material for details) with initial conditions given by an unstabletravelling wave perturbed along the direction of the Floquet mode causing the instability. Thetypical dynamical scenario is shown in figure 6 for the unstable wave with c = 1.034. Theeigenvalue spectrum of this solution is shown in figure 4. One can see that linear waves arecontinuously being created, and the solitary wave degrades with time.

Turning now to our second example, we consider the model studied in [37,41–43], the α-FPUlattice [14–16] with long-range Kac–Baker interactions that have exponentially decaying kernel.This corresponds to (6.1) with

V(r) = r2

2− r3

3, Λ(m) = ρ(eγ − 1) e−γ |m|(1 − δm,0). (6.10)

Long-range interactions of this type have been argued to be of relevance for modelling Coulombinteractions in DNA [37]. As shown in [37], when

ρ1 <ρ < ρ2,



21


.........................................................

1.5 1.6 1.7 1.8 1.9 2.00

10

20

30

40

50

60

70

80

H(c

)

c

1.68 1.69 1.70 1.71 1.72 1.73 1.7426.0

26.5

27.0

27.5

28.0

28.5

1.690 1.695 1.700 1.705 1.710 1.715 1.7200

0.5

1.0

1.5

2.0

l(×

10–3

)

c

Figure7. Travelling solitarywaves in theα-FPUpotentialwithKac–Baker long-range interactions forγ = 0.18 andρ = 0.02.(a) The energy dependence on the velocity of the waves, with H′(c)> 0 implying (spectral) stability, and H′(c)< 0 implyinginstability. The inset zooms in on the portion of the curve where H′(c) changes sign. Panel (b) confirms this by showing themaximum real eigenvalue obtained by diagonalizing the linearization operatorL (dots) and transforming the relevant Floquetmultiplierμ into a corresponding eigenvalue (for comparison) via the relationλ= log(μ) (solid curve). The vertical lines in (b)indicate the values of c where monotonicity of H(c) changes. (Online version in colour.)

where

ρ1 ≈

⎧⎪⎨⎪⎩

0.23γ 4

γ 21 − γ 2

, γ < γ1

∞, γ ≥ γ1

and ρ2 ≈

⎧⎪⎨⎪⎩

38

γ 4

γ 22 − γ 2

, γ < γ2

∞, γ ≥ γ2,(6.11)

and γ1 = 0.25 and γ2 = 0.16, the energy H(c) of the supersonic solitary travelling waves in thissystem is non-monotone so that the curve presents a local maximum and a local minimum.Clearly, for this to hold it is necessary that γ < γ1. The energy of the waves tends to zero when capproaches the sound velocity

cs =√

1 + ρ1 + exp(−γ )

(1 − exp(−γ ))2 .

In what follows, we choose γ = 0.18 and ρ = 0.02, which places the system in the regime withnon-monotone H(c) and yields cs = 1.5338. The profiles of solitary travelling waves with differentvelocities c are depicted in figure 2b. The resulting non-monotone H(c) curve is shown in figure 7a.In particular, H′(c) is negative for c0,max < c< c0,min, where c0,max = 1.6958 and c0,min = 1.7185denote, as in the previous example, the velocity values at which H(c) reaches a local maximumand a local minimum, respectively. Very close to this interval of velocities, an eigenvalue of theoperator L crosses through λ= 0 (dots in figure 7b) and a Floquet mode pair crosses (1, 0) (solidline in figure 7b). As discussed above, there are numerical effects caused by the finite length l of thelattice; as a result, the interval of instabilities observed by means of the spectral stability analysis is1.6971< c< 1.7137 for l = 400 and 1.6964< c< 1.7161 for l = 800 (in both cases, we used�ξ = 0.1).Clearly, however, the instability interval approaches the interval for which H′(c)< 0. Figure 8shows the spectrum of L and the Floquet mode spectrum for an unstable travelling wave withvelocity c = 1.7 in this interval.

The connection with the theoretical analysis of theorem 5.3 is presented in figure 9, wherethe dependence of λ2 with respect to c close to cl

0 is shown in the vicinity of both bifurcations,while table 2 compares the corresponding numerical values of the slope β and its theoreticalprediction (6.9). For doing such comparisons, numerical computation has been performed usingl = 800 and �ξ = 0.1. There is a mismatch of approximately 9% and approximately 5% at the firstand second bifurcation, respectively. The higher mismatch at the first bifurcation is a consequenceof the slower decay of the solitary wave in that velocity interval. One should also keep in mind the



22


.........................................................

–3 –2 –1 0 1 2 3–3

–2

–1

0

1

2

3Im

(l)

Re (l) (×10–3)

–2 0 2×10–3

–40

–20

0

20

40

–1.0 –0.5 0 0.5 1.0

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

Im(m

)

Re (m)

0.998 1.000 1.002–4

–2

0

2

4

(×10

–3)

(b)(a)

Figure 8. Stability analysis of the unstable travelling wave with velocity c = 1.7 for the α-FPU potential with Kac–Bakerlong-range interactions forγ = 0.18 andρ = 0.02. (a) The spectrum of the linearization operatorL for Im (λ) ∈ (−π ,π );the inset corresponds to the full spectrum. In (b), the Floquet multipliers spectrum is shown. (Online version in colour.)

1.69640 1.69641 1.69642 1.69643 1.69644 1.69645–10

–8

–6

–4

–2

0

2

4

6

8

l2 (×

10–9

)

c1.71612 1.71613 1.71614 1.71615 1.71616 1.71617

–3

–2

–1

0

1

2

3

4

5

c

(b)(a)

Figure 9. Dependence of the squared linearization eigenvalueλ2 on c for solitary wave solutions for theα-FPU potential withKac–Baker long-range interactions for γ = 0.18, ρ = 0.02 in the vicinity of bifurcations at c0,max (a) and c0,min (b). The solidline in each panel corresponds to the best linear fit from which the corresponding numerical value of β is obtained. (Onlineversion in colour.)

Table 2. Same as table 1 but for theα-FPU model with Kac–Baker long-range interactions for γ = 0.18 andρ = 0.02.

c0 cl0 H′′(c0) α1 βt β

1.6958 1.69642 −3.4325 × 104 −1.0922 × 108 3.1428 × 10−4 2.8617 × 10−4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.7185 1.71615 5.9857 × 103 −6.0313 × 107 −1.0505 × 10−4 −0.9924 × 10−4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

potential contribution from the essential spectrum (which is expected to be of higher order) in theframework with unweighted spaces considered here. Generally, however, we find the agreementbetween theoretical and numerical results to be very good, which makes it reasonable to neglectthe effects of the essential spectrum at the current order.

Finally, we discuss the dynamics of unstable travelling waves perturbed along the direction ofthe Floquet mode causing the instability. As a typical example, we consider the unstable solution



23


.........................................................

t/T

n200 400 600 800 1000 1200 1400

n0 200 400 600 800 1000 1200 14000

100

200

300

400

500

600

700

800

–1.2

–1.0

–0.8

–0.6

–0.4

–0.2

0

r n (t

= 3

00T

)

(b)(a)

Figure 10. Evolution of the unstable travelling wave with velocity c = 1.7 for theα-FPUmodel with Kac–Baker interaction forγ = 0.18 and ρ = 0.02. The spectrum of the corresponding linearization operator is shown in figure 8. (a) The space–timeevolution dynamics of the strains; (b) the strain profile at t = 300T , where T = 1/c. (Online version in colour.)

with velocity c = 1.7, whose spectrum is shown in figure 8. One can see that a linear wave movingat the speed of sound is expelled from the supersonic solitary wave. Subsequently, the solitarywave moves with a velocity c = 1.7478, which corresponds to a stable wave (note that H′(c)> 0 atthis value, according to the inset in figure 7a), as shown in figure 10.

7. Concluding remarks and future challengesIn this work, we have revisited the stability problem of solitary travelling waves in infinite-dimensional nonlinear Hamiltonian lattices. We considered a class of problems where such wavesexist for a range of velocities, with our examples predominantly drawn from the pool of FPU-type models. We have illustrated the useful stability criterion originally put forth in the workof Friesecke & Pego [22] and have extended it through a systematic analysis of the associatedlinearized problem. In all examples that we have explored, a decreasing H(c) was associatedwith instability, and an increasing one with stability. The formulation of the stability problemwas considered from the perspective of the co-travelling frame steady state, whose linearizationpossesses a pair of eigenvalues moving from imaginary to real values as the instability thresholdis traversed. It was also explored from the perspective of a periodic orbit calculation with a pairof Floquet multipliers (directly connected to the above eigenvalues) transitioning from the unitcircle to the real axis, as the wave becomes unstable. The use of time-translation invariance andassociated eigenvector and generalized eigenvectors provided a framework for the analyticalcalculation of the relevant eigenvalues, which is crucial towards identifying the growth rate ofthe associated instability near the critical point at which the travelling wave becomes unstable.Connections were also established both with the stability theory of discrete breathers (where anintimately connected criterion exists [40], provided that we replace the travelling wave velocitywith the breather frequency) and with the more general theory of [49] (see §V of the electronicsupplementary material).

Nevertheless, there are numerous open problems in this direction that merit furtherinvestigation in the near future. Below we mention a few representative questions.

— We brought to bear perturbation techniques to study the motion of the zero eigenvaluein unweighted spaces and ignored the contribution from the essential spectrum, whichbased on the numerical results (at least in the finite domains considered) appears toosmall to affect our leading-order approximations. Nevertheless, it would certainly be



24


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useful to investigate the exact order of the contribution from the essential spectrum, as astep towards a full understanding of the perturbation mechanism.

— In all of the examples we considered, the parameter α1 was found to be negative, yieldingtransition from stability to instability when the sign of H′(c) changes from positive tonegative. Are there examples of interest where α1 is positive and hence these stabilityconclusions can be reversed?

— We assumed the existence of a family of solitary travelling waves parametrized byvelocity in some continuous range. However, there are other examples, notably, e.g.of the Klein–Gordon variety, where travelling waves with vanishing tails are isolatedin terms of their velocities. In such cases, travelling waves may exist as members of afamily of solutions with non-vanishing tails [50]. It is important to understand whetherour criterion can still be applied in these cases and more broadly in the setting oftravelling ‘nanoptera’, where the non-vanishing nature of the tails would not allow theconsideration of the spaces and assumptions made in our analysis.

— Another question of interest concerns the case of dissipative gradient systems. It isworthwhile to examine whether in such settings there may exist a suitable modificationor extension of the formulation presented here that yields an associated stability criterion,perhaps based on the free energy.

Some of these directions are presently under consideration and will be reported in futurepublications.

Data accessibility. There were no experimental data collected for this publication. The algorithms and set-upsfor numerical computing were explained in §6 and section IV, electronic supplementary material. The codesused to produce numerical results are available from the authors upon request. Relevant requests may beaddressed to: [email protected]’ contributions. P.G.K. and A.V. conceived the overall project and designed the research. H.X. wasprincipally responsible for the mathematical analysis and J.C.M. for the numerical computations. All theauthors discussed the results extensively, and all of them contributed to the writing of and approved themanuscript.Competing interests. The authors declare that they have no competing interests.Funding. J.C.-M. is thankful for financial support from MAT2016-79866-R project (AEI/FEDER, UE). A.V.acknowledges support by the U.S. National Science Foundation through the grant no. DMS-1506904.P.G.K. gratefully acknowledges support from the Alexander von Humboldt Foundation, the Greek DiasporaFellowship Program, the US-NSF under grant no. PHY-1602994. as well as the ERC under FP7, Marie CurieActions, People, International Research Staff Exchange Scheme (IRSES-605096).Acknowledgements. The authors thank Dmitry Pelinovsky for fruitful discussions.

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