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On the energy exchange between resonant modes in nonlinear Schrödinger equations


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On the energy exchange between resonant modes in nonlinear Schrödinger equations

Echange d’énergie entre modes résonants dans une équation de Schrödinger non linéaire cubique

Benoît Grébert


, Carlos Villegas-Blas


aLaboratoire de Mathématiques Jean Leray UMR 6629, Université de Nantes, 2, rue de la Houssinière, 44322 Nantes Cedex 3, France bUniversidad Nacional Autonoma de México, Instituto de Matemáticas, Unidad Cuernavaca, Mexico

Received 23 July 2010; received in revised form 22 November 2010; accepted 22 November 2010 Available online 1 December 2010


We consider the nonlinear Schrödinger equation

t= −ψxx±2 cos 2x|ψ|2ψ, xS1, t∈R

and we prove that the solution of this equation, with small initial datumψ (0, x)=ε(Aexp(ix)+Bexp(−ix)), will periodically exchange energy between the Fourier modeseixandeixas soon asA2=B2. This beating effect is described up to time of order ε5/2while the frequency is of orderε2. We also discuss some generalizations.

©2010 Elsevier Masson SAS. All rights reserved.


Nous considérons l’équation de Schödinger non linéaire

t= −ψxx±2 cos 2x|ψ|2ψ, xS1, t∈R

et nous montrons la solution de cette équation ayant pour donnée initialeψ (0, x)=ε(Aexp(ix)+Bexp(−ix))avecεpetit, va échanger périodiquement de l’énergie entre les modes de Fouriereixeteixdès queA2=B2. Cet effet de battement, dont la période est de l’ordre deε2, est mis en évidence pour des temps de l’ordre deε5/2. Nous présentons aussi quelques généralisa- tions.

©2010 Elsevier Masson SAS. All rights reserved.

MSC:37K45; 35Q55; 35B34; 35B35

Keywords:Normal form; Nonlinear Schrödinger equation; Resonances; Beating effect

B. Grébert supported in part by the grant ANR-06-BLAN-0063. C. Villegas-Blas supported in part by PAPIIT-UNAM IN109610.

* Corresponding author.

E-mail addresses:benoit.grebert@univ-nantes.fr (B. Grébert), villegas@matcuer.unam.mx (C. Villegas-Blas).

0294-1449/$ – see front matter ©2010 Elsevier Masson SAS. All rights reserved.



Mots-clés :Forme normale ; Equation de Schrödinger non linéaire ; Résonances ; Échange d’énergie

1. Introduction

Let us consider the nonlinear Schrödinger equation (NLS) on the circleS1

t= −ψxx+Vψ+g(x, ψ,ψ )¯ (1.1)

whereVΨ denotes the convolution function between a potentialV :S1→Rand the functionΨ. The nonlinear termgis Hamiltonian in the sense thatg=ψ¯GwithGanalytic with respect to its three variables andG(x, z,z)¯ ∈R.

We further assume thatGis globally at least of order three in(z,z)¯ at the origin in such a way thatgis effectively a nonlinear term.

The Fourier basis exp(ij x), j ∈Z provides an orthonormal basis forL2(S1)in which the linear operator A=

∂x22 +V∗is diagonal. The corresponding eigenvalues ofAare given by the real numbersωj=j2+ ˆV (j ),j∈Z, whereV (j )ˆ are the Fourier coefficients ofV:

V (j )ˆ = π


V (x)exp(−ij x) dx (1.2)

wheredxdenotes the normalized Lebesgue measure onS1.

Under a non-resonant condition on the frequenciesωj, it has been established by D. Bambusi, B. Grébert [1] (see also [3]) that, given a numberr3, for small initial data in theHs norm,1sayψ0s =εwiths large enough, the solution to the NLS equation (1.1) remains small in the same Sobolev norm,ψ (t )s2εfor a large period of time,

|t|εr. Furthermore the actionsIjξjηj,j∈Z, are almost invariant during the same period of time. In [1], it is also proved that in the more natural case of a multiplicative potential,

t= −ψxx+V ψ+g(x, ψ,ψ ),¯

which corresponds to an asymptotically resonant case,ωjωj when |j| → ∞, we can generically impose non- resonant conditions on the frequencies {ω|j|}in such a way that the generalized actionsJj =Ij+Ij are almost conserved quantities. A priori nothing preventsIj andIj from interacting. In this article, we exhibit a nonlinearity gthat incites this interaction and especially a beating effect. This is a nonlinear effect which is a consequence of the resonances in the linear part. We will see that not all nonlinearities incite the beating. For instance ifgdoes not depend onx there is no beating for a long time. Typically, to obtain an interaction between the modej and the mode−j, the nonlinearitygmust contain oscillations of frequency 2j or a multiple of 2j, i.e.gmust depend onxas cos 2kj xor sin 2kj xfor somek1. In what follows, we will focus on the totally resonant caseV =0 and a cubic nonlinearity g=2 cos 2x|ψ|2ψ. Namely, we consider the following initial value problem:

t= −ψxx±2 cos 2x|ψ|2ψ, xS1, t∈R,

ψ (0, x)=ε(Aexp(ix)+Bexp(−ix)) (1.3)

whereεis a small parameter andA, B∈Cwith|A|2+ |B|2=1. By classical arguments based on the conservation of the energy and Sobolev embeddings, this problem has a unique global solution in all the Sobolev spacesHs fors1 and even fors0 using more refined technics (see for instance [2, Chapter 5]). Our result precises the behavior of the solution for time of orderε5/2. We write the solutionψin Fourier modes:ψ (t, x)=

j∈Zξj(t )eij x. Theorem 1.1.Forεsmall enough we have for|t|ε5/2


ξ1(t )2+ξ1(t )2=ε2+O ε7/2 ,

ξ1(t )2ξ1(t )2= ±ε2A2B2sin 2ε2t+δ

+O ε7/2

where the phaseδis some explicit constant depending onAandBgiven by(3.1).

1 HereHsdenotes the standard Sobolev Hilbert space onS1and · sits associated norm.


In other words, the first estimate says that all the energy remains concentrated on the two Fourier modes+1 and

−1 and the second estimate says that, as soon asA2=B2, there is energy exchange, namely a beating effect, between these two modes. For instance whenψ (0, x)=ε(cosx+sinx)and thusA=12i andB=1+2i, we observe a total beating for|t|ε5/2:

ξ1(t )2=ε21±sin 2ε2t

2 +O ε7/2

, ξ1(t )2=ε21∓sin 2ε2t

2 +O ε7/2 .

On the contrary whenA= ±Bno exchange of energy is observed for|t|ε5/2. Notice that the sign in front of the nonlinearity does not affect the phenomena.

That the principal term ofgbe cubic is certainly not necessary but convenient for the calculations. We see that the period of the beating depends on the nonlinearity but also on the value of all the initial actions (cf. Section 4).

The paper is organized as follows: in Section 2 we prove a normal form result for Eq. (1.3) that allows us to reduce the initial problem to the study of a finite dimensional classical system. The main theorem is proved in Section 3.

Section 4 is devoted to generalizations and comments.

2. The normal form

Let us expandψandψ¯ in Fourier modes:

ψ (t, x)=


ξj(t )eij x, ψ (t, x)¯ =


ηj(t )eij x.

In this Fourier setting Eq. (1.3) reads as an infinite Hamiltonian system ˙j=j2ξj+∂η∂Pj j∈Z,

˙j=j2ηj+∂ξ∂Pj j∈Z (2.1)

where the perturbation term is given by P (ξ, η)= ±2


cos 2xψ (x)4dx

= ±

j,l∈Z2 M(j,l)2

ξj1ξj2ηl1ηl2. (2.2)

andM(j, l)=j1+j2l1l2denotes the momentum of the multi-index(j, l)∈Z4. In the sequel we will develop the calculus with± = +but all remains true, mutatis mutandis, with the minus sign.

Since the regularity is not an issue in this work, we will work in the phase space (ρ0) Aρ=

(ξ, η)1(Z)×1(Z)(ξ, η)



eρ|j| |ξj| + |ηj|


that we endow with the canonical symplectic structure−i

jjj. Notice that this Fourier space corresponds to functionsψ (z)analytic on a strip|Imz|< ρaround the real axis.

According to this symplectic structure, the Poisson bracket between two functionsf andgof(ξ, η)is defined by {f, g} = −i









In particular, if(ξ(t ), η(t ))is a solution of (2.1) andF is some regular Hamiltonian function, we have d

dtF ξ(t ), η(t )

= {F, H} ξ(t ), η(t ) whereH=H0+P =

j∈Zj2ξjηj+P is the total Hamiltonian of the system.

We denote byBρ(r)the ball of radiusrcentered at the origin inAρ. In the next proposition we put the Hamiltonian Hin normal form up to order 4:


Proposition 2.1.There exists a canonical change of variableτ fromBρ(ε)intoBρ(2ε)withεsmall enough such that

Hτ =H0+Z4,1+Z4,2+Z4,3+R6 (2.3)


(i) H0(ξ, η)=


(ii) Z4=Z4,1+Z4,2+Z4,3is the(resonant)normal form at order4, i.e.Z4is a polynomial of order4satisfying {H0, Z4} =0.

(iii) Z4,1(ξ, η)=2(ξ1η1+ξ1η1)(2

p∈Z,p=1,1ξpηp+1η1+ξ1η1))is the effective Hamiltonian at order4.

(iv) Z4,2(ξ, η)=4(ξ2ξ1η2η1+ξ2ξ1η2η1).

(v) Z4,3contains all the terms inZ4involving at most one mode of index1or−1.

(vi) R6is the remainder of order6, i.e. a Hamiltonian satisfyingXR6(z)ρCz5ρforz=(ξ, η)Bρ(ε).

(vii) τ is close to the identity:there exists a constantCρsuch thatτ (z)zρCρz2ρfor allzBρ(ε).

Proof. The proof uses the classical Birkhoff normal form procedure (see for instance [4] in a finite dimensional setting or [3] in the infinite dimensional one). Since the free frequencies are the square of the integers, we are in a totally resonant case and we are not facing the small denominator problems: a linear combination of integer numbers with integer coefficients is exactly 0 or its modulus equals at least 1. For convenience of the reader, we briefly recall the procedure. Let us searchτ as time one flow ofχa polynomial Hamiltonian of order 4,


j,l∈Z2 M(j,l)2


LetF be a Hamiltonian, one has by using the Taylor expansion ofFΦχt betweent=0 andt=1:

Fτ =F + {F, χ} + 1 0

(1t )

{F, χ}, χ

Φχt dt.

Applying this formula toH=H0+P we get Fτ =H0+P+ {H0, χ} + {P , χ} +

1 0

(1t )

{H, χ}, χ

Φχt dt.

Therefore in order to obtainHτ =H0+Z4+R6we need to solve the homological equation

{χ , H0} +Z4=P (2.4)

and then we define R6= {P , χ} +

1 0

(1t )

{H, χ}, χ

Φχt dt. (2.5)

Forj, l∈Z2we define the associated divisor by Ω(j, l)=j12+j22l12l22.

The homological equation (2.4) is solved by defining


j,l∈Z2 M(j,l)2,Ω(j,l)=0


iΩ(j, l)ξj1ξj2ηl1ηl2 (2.6)



j,∈Z2 M(j,)2,Ω(j,)=0

ξj1ξj2η1η2. (2.7)


At this stage we remark that any formal polynomial


j,l∈Z2 M(j,)2


is well defined and continuous (and thus analytic) on Aρ as soon as theaj form a bounded family. Namely, if

|aj,|Mfor allj, ∈Z2then Q(ξ, η)M

16 (ξ, η)4


16(ξ, η)4


Furthermore the associated vector field is bounded (and thus smooth) fromAρ toAρ, namely XQ(ξ, η)



eρ|k| ∂P

∂ξk +






j1,1,2∈Z M(j1,k,1,2)2

|ξj1η1η2| + |ξ1ξ2ηj1|







2 e(ξ, η)3

ρ (2.8)

where we used,M(j1, k, 1, 2)= ±2⇒ |k|2+ |j1| + |1| + |2|.

Since there are no small divisors in this resonant case, Z4 andχ are well defined on Aρ and by construction Z4 satisfies (ii). On the other hand, sinceχ is homogeneous of order 4, forεsufficiently small, the time one flow generated byχmaps the ballBρ(ε)into the ballBρ(2ε)and is close to the identity in the sense of assertion (vii).

ConcerningR6, by construction it is a Hamiltonian function which is of order at least 6. To obtain assertion (vi) it remains to prove that the vector fieldXR6 is smooth fromBρ(ε)intoAρ in such a way we can Taylor expandXR6 at the origin. This is clear for the first term of (2.5),{P , χ}. For the second, notice that{H, χ} =Z4P+ {P , χ}which is a polynomial onAρ having bounded coefficients2and the same is true forQ= {{H, χ}, χ}. Therefore, in view of the previous paragraph,XQis smooth. Now, since forεsmall enoughΦχt maps smoothly the ballBρ(ε)into the ball Bρ(2ε)for all 0t1, we conclude that1

0(1t ){{H, χ}, χ} ◦Φχt dthas a smooth vector field.

We are now interested in describing more explicitly the terms appearing in the expression (2.7) on base of the number of times thatξ±1orη±1appear as a factor. Let us define the resonant set


(j1, j2, j3, j4)∈Z4j1+j2j3j4= ±2 andj12+j22j32j42=0

(2.9) in such a way (2.7) reads



ξj1ξj2ηj3ηj4. (2.10)

Note that it is enough to deal with the casej1+j2j3j4=2 because the other casej1+j2j3j4= −2 can be obtained by changing the signs ofj1, j2, j3, j4. Thus let us study two cases:j1=1 orj1= −1.

In the first case we havej2=j3+j4+1 and therefore the quadratic condition implies(1+j3)(1+j4)=0. Thus we conclude that(j1, j2, j3, j4)=(1, p,−1, p)or(j1, j2, j3, j4)=(1, p, p,−1). Thus whenj1=1, the only terms appearing in the expression forZ4in Eq. (2.10) must have the formξ1ξpη1ηporξ1ξpηpη1.

In the second case, j = −1, we have j2 =j3 +j4+3 which in turn implies 5+3(j3+j4)+j3j4 =0.

By looking at the graph of the function f (x)= −53++3xx we find that the only acceptable cases appearing in Eq. (2.10) are(j1, j2, j3, j4)=(−1,1,−1,−1),(−1,2,−2,1),(−1,2,1,−2),(−1,−8,−4,−7),(−1,−8,−7,−4) or(−1,−7,−5,−5).

2 Note that this is not true forH0.


Noting that the resonant setRis invariant under permutations ofj1withj2orj3withj4we obtain that Z4=




1η1+ξ1η1) +4(ξ2ξ1η2η1+ξ2ξ1η2η1)

+4(ξ1ξ8η7η4+ξ1ξ8η7η4)+2(ξ1ξ7η5η5+ξ1ξ7η5η5)+ ˜Z4

whereZ˜4is equal to the sum of terms of the formξj1ξj2ηj3ηj4with(j1, j2, j3, j4)Rand satisfying the condition

|jk| =1, for allk=1,2,3,4. 2 3. Dynamical consequences

We denote byIp=ξpηp,p∈Z, the actions, byJp=Ip+Ipforp∈N\ {0}andJ0=I0, the generalized actions and byJ=

p∈ZIp. Notice that, when the initial condition0, η0)of the Hamiltonian system (2.1) satisfiesη0= ¯ξ0 (and this is actually the case whenξ0andη0are the sequences of Fourier coefficients ofψ0,ψ¯0) respectively, this reality property is conserved i.e.η(t )= ¯ξ (t )for allt. As a consequence all the quantitiesIp,JpandJ are real and positive. We also recall that theL2norm ofψ (t,·)is conserved by the NLS equation and thusJ (t )=J (0)for allt.

To begin with we establish that apart fromI1andI1the others actions are almost constant:

Lemma 3.1.Letψ (t,·)=

k∈Zξk(t )eikxbe the solution of (1.3)then forεsmall enough and|t|ε5/2, Ip(t )ε4 forp∈Z\ {−1,+1}.

Proof. Using Proposition 2.1 we have for allp∈N J˙p= {Jp, H} = {Jp, Z4,3} + {Jp, R6}

since by elementary calculations{Jp, Z4,1} = {Jp, Z4,2} =0.

LetTεbe the maximal time such that for all|t|Tε Jp(t )ε4, forp=1.

SinceJ (t )=J (0)=ε2, we get forεsmall enough and|t|Tε

ξ1(t ), ξ1(t ), η1(t ),η1(t )ε whileξp(t ),ηp(t )ε2 forp= ±1.

Therefore, from the definition ofZ4,3andR6we deduce that for|t|Tε {Jp, R6} =O ε7

, {Jp, Z4,3} =O ε7

forp=1, and thus there existsC >0 such that for any|t|Tε

| ˙Jp|7 forp=1.

Taking into account thatJp(0)=0, we obtain Jp(t )C|t|ε7 forp=1

for all|t|Tεand we conclude by a classical bootstrap argument thatTεC1ε3ε5/2forεsmall enough. 2 In order to prove Theorem 1.1 let us define some quadratic Hamiltonian functions(p∈Z):

Mp:=ξpηpξpηp, Jp:=ξpηp+ξpηp, Lp:=i(ξpηpξpηp), Kp:=ξpηp+ξpηp. One computes

M˙1= {M1, H} = {M1, Z4,1+Z4,2} + {M1, Z4,3+R6}

=2J L1+L1K2K1L2+ {M1, Z4,3+R6}


and using Lemma 3.1 we get that for|t|ε5/2, M˙1=2J L1+O ε6

and in the same way we verify L˙1= −2J M1+O ε6


We can now compute the solution of the associated linear ODE (recall thatJis an invariant of the motion) M˙1=2J L1,

L˙1= −2J M1

to conclude that for|t|ε5/2

M1(t )=M1(0)cos 2J t+L1(0)sin 2J (0)t+O ε7/2 .

In Theorem 1.1 we have chosenψ (0, x)=ε(Aexp(ix)+Bexp(−ix))which corresponds toξ1(0)=Aε,η1(0)=Aε, ξ1(0)=andη1(0)=Bε. ThereforeJ=J1(0)=(|A|2+ |B|22=ε2,M1(0)=(|A|2− |B|22andL1(0)=

−2 Im{AB}ε2which implies M1(t )= |A|2− |B|2

ε2cos 2 |A|2+ |B|2 ε2t

−2 Im{AB}ε2sin 2ε2t

+O ε9/4

=ε2A2B2sin 2ε2t+δ

+O ε9/4 withδdetermined by the equations:

sinδ= |A|2− |B|2A2B2, cosδ= −2 Im{AB}/A2B2.

(3.1) We finally remark that choosing the minus sign in front of the nonlinearity will lead to the following linear system

M˙1= −2J L1, L˙1=2J M1

which again gives the desired result.

4. Generalizations and comments

– The same type of result remains true when we add a higher order term to the nonlinearity, i.e. considering the equation

t= −ψxx±2 cos 2x|ψ|2ψ+O |ψ|4

, xS1, t∈R.

– We can prove a similar result when changing the nonlinearity in such a way that we still privilege the modes 1 and−1. The game is to conserve an effective Hamiltonian at order 4 (see Proposition 2.1). For instance 2 cos 2x can be replaced byacos 2x+bsin 2xbut not by cos 4xwhich generates an effective Hamiltonian only at order 6.

We can also choose to privilege another couple of modesp and−pby choosingacos 2px+bsin 2px. In that case we have also to adapt the initial datum.

– If we choose a nonlinearity that does not depend onx, for instance the standard cubic nonlinearityg= ±|ψ|2ψ, then we can prove that there is no beating effect between any modes for|t|ε3sinceZ4only depends on the actions in that case (independently of the sign in front of the nonlinearity).

– The beating frequency, namely 2J, depends on all the modes initially excited. For instance if ψ0 = ε(cosx+sinx)+ε2cosqx(q=1) then we can still prove that there is no energy exchanges between the modep, for|p| =1, and modes 1 and−1, that there is the same beating effect between modes 1 and−1, nevertheless the beating frequency is slightly changed: 2J=2ε2+ε4.

– As stated in the introduction, when adding a linear potential – a multiplicative one or a convolution one – we can choose the potential in order to avoid resonances between the different blocks of modes pand−p(see [1] for multiplicative potentials or [3] for convolution potentials). In that case the same result can be proved and actually in an easier way since we avoid exchanges between the blocks for arbitrary long time.



The authors acknowledge an anonymous referee for useful comments that allowed to improve the estimates.

The first author is glad to thank the hospitality of the Mathematical Institute of Cuernavaca (UNAM, Mexico) where this work was initiated.


[1] D. Bambusi, B. Grébert, Birkhoff normal form for PDEs with tame modulus, Duke Math. J. 135 (2006) 507–567.

[2] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Amer. Math. Soc. Colloq. Publ., vol. 46, Amer. Math. Soc., Providence, RI, 1999.

[3] B. Grébert, Birkhoff normal form and Hamiltonian PDEs, in: Partial Differential Equations and Applications, in: Sémin. Congr., vol. 15, Soc.

Math. France, Paris, 2007, pp. 1–46.

[4] J. Moser, Lectures on Hamiltonian systems, Mem. Amer. Math. Soc. 81 (1968) 1–60.


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