Page 1

arXiv:0810.0735v9 [math.AP] 20 Aug 2009

SOLITON DYNAMICS FOR CNLS SYSTEMS WITH POTENTIALS

EUGENIO MONTEFUSCO, BENEDETTA PELLACCI, AND MARCO SQUASSINA

Abstract. The semiclassical limit of a weakly coupled nonlinear focusing Schr¨ odinger sys-

tem in presence of a nonconstant potential is studied. The initial data is of the form (u1,u2)

with ui= ri

?x−˜ x

(φ1,φ2) will been shown to have, locally in time, the form (r1

where (x(t),ξ(t)) is the solution of the Hamiltonian system ˙ x(t) = ξ(t),˙ξ(t) = −∇V (x(t))

with x(0) = ˜ x and ξ(0) =˜ξ.

ε

?e

i

εx·˜ξ, where (r1,r2) is a real ground state solution, belonging to a suit-

able class, of an associated autonomous elliptic system. For ε sufficiently small, the solution

?x−x(t)

ε

?e

i

εx·ξ(t),r2

?x−x(t)

ε

?e

i

εx·ξ(t)),

1. Introduction and main result

1.1. Introduction. In recent years much interest has been devoted to the study of systems

of weakly coupled nonlinear Schr¨ odinger equations. This interest is motivated by many

physical experiments especially in nonlinear optics and in the theory of Bose-Einstein con-

densates (see e.g. [1, 17, 24, 26]). Existence results of ground and bound states solutions have

been obtained by different authors (see e.g. [3, 5, 13, 21, 22, 30]). A very interesting aspect

regards the dynamics, in the semiclassical limit, of a general solution, that is to consider

the nonlinear Schr¨ odinger system

with 0 < p < 2/N, N ≥ 1 and β > 0 is a constant modeling the birefringence effect of the

material. The potential V (x) is a regular function in RNmodeling the action of external

forces (see (1.11)), φi: R+× RN→ C are complex valued functions and ε > 0 is a small

parameter playing the rˆ ole of Planck’s constant. The task to be tackled with respect to this

system is to recover the full dynamics of a solution (φε

galileian motion for the parameter ε sufficiently small. Since the famous papers [2, 14, 16],

a large amount of work has been dedicated to this study in the case of a single Schr¨ odinger

equation and for a special class of solutions, namely standing wave solutions (see [4] and the

(1.1)

iε∂tφ1+ε2

2∆φ1− V (x)φ1+ φ1(|φ1|2p+ β|φ2|p+1|φ1|p−1) = 0

iε∂tφ2+ε2

2∆φ2− V (x)φ2+ φ2(|φ2|2p+ β|φ1|p+1|φ2|p−1) = 0

φ1(0,x) = φ0

φ2(0,x) = φ0

in RN× R+,

in RN× R+,

1(x)

2(x),

1,φε

2) as a point particle subjected to

2000 Mathematics Subject Classification. 34B18, 34G20, 35Q55.

Key words and phrases. Weakly coupled nonlinear Schr¨ odinger systems, concentration phenomena, semi-

classical limit, orbital stability of ground states, soliton dynamics.

The first and the second author are supported by the MIUR national research project “Variational

Methods and Nonlinear Differential Equations”, while the third author is supported by the 2007 MIUR

national research project “Variational and Topological Methods in the Study of Nonlinear Phenomena”.

1

Page 2

2E. MONTEFUSCO, B. PELLACCI, AND M. SQUASSINA

references therein). When considering this particular kind of solutions one is naturally lead

to study the following elliptic system corresponding to the physically relevant case p = 1

(that is Kerr nonlinearities)

(1.2)

?

−ε2∆u + V (x)u = u3+ βv2u

−ε2∆v + V (x)v = v3+ βu2v

in RN,

in RN,

so that the analysis reduces to the study of the asymptotic behavior of solutions of an elliptic

system. The concentration of a least energy solution around the local minima (possibly

degenerate) of the potential V has been studied in [27], where some sufficient and necessary

conditions have been established. To our knowledge the semiclassical dynamics of different

kinds of solutions of a single Schr¨ odinger equation has been tackled in the series of papers [7,

18, 19] (see also [6] for recent developments on the long term soliton dynamics), assuming

that the initial datum is of the form r((x − ˜ x)/ε)e

solution of an associated elliptic problem (see equation (1.8)) and ˜ x,˜ξ ∈ RN. This choice of

initial data corresponds to the study of a different situation from the previous one. Indeed,

it is taken into consideration the semiclassical dynamics of ground state solutions of the

autonomous elliptic equation once the action of external forces occurs. In these papers it is

proved that the solution is approximated by the ground state r–up to translations and phase

changes–and the translations and phase changes are precisely related with the solution of

a Newtonian system in RNgoverned by the gradient of the potential V . Here we want to

recover similar results for system (1.1) taking as initial data

i

εx·˜ξ, where r is the unique ground state

(1.3)φ0

1(x) = r1

?x − ˜ x

ε

?

e

i

εx·˜ξ,φ0

2(x) = r2

?x − ˜ x

ε

?

e

i

εx·˜ξ,

where the vector R = (r1,r2) is a suitable ground state (see Definition 1.3) of the associated

elliptic system

(E)

−1

−1

2∆r1+ r1= r1(|r1|2p+ β|r2|p+1|r1|p−1)

2∆r2+ r2= r2(|r2|2p+ β|r1|p+1|r2|p−1)

in RN,

in RN.

When studying the dynamics of systems some new difficulties can arise. First of all, we

have to take into account that, up to now, it is still not known if a uniqueness result (up

to translations in RN) for real ground state solutions of (E) holds. This is expected, at

least in the case where β > 1. Besides, also nondegeneracy properties (in the sense provided

in [12, 28]) are proved in some particular cases [12, 28]. These obstacles lead us to restrict the

set of admissible ground state solutions we will take into consideration (see Definition 1.3)

in the study of soliton dynamics.

Our first main result (Theorem 1.5) will give the desired asymptotic behaviour. Indeed,

we will show that a solution which starts from (1.3) (for a suitable ground state R) will

remain close to the set of ground state solutions, up to translations and phase rotations.

Furthermore, in the second result (Theorem 1.9), we will prove that the mass densities

associated with the solution φiconverge–in the dual space of C2(RN)×C2(RN)–to the delta

Page 3

SOLITON DYNAMICS FOR CNLS SYSTEMS WITH POTENTIALS3

measure with mass given by ?ri?L2 and concentrated along x(t), solution to the (driving)

Newtonian differential equation

(1.4)¨ x(t) = −∇V (x(t)),x(0) = ˜ x, ˙ x(0) =˜ξ

where ˜ x and˜ξ are fixed in the initial data of (1.1). A similar result for each single component

of the momentum density is lost as a consequence of the birefringence effect. However, we

can afford the desired result for a balance on the total momentum density. This shows that–

in the semiclassical regime–the solution moves as a point particle under the galileian law

given by the Hamiltonian system (1.4). In the case of V constant our statements are related

with the results obtained, by linearization procedure, in [31] for the single equation. Here,

by a different approach, we show that (1.4) gives a modulation equation for the solution

generated by the initial data (1.3). Although we cannot predict the shape of the solution,

we know that the dynamic of the mass center is described by (1.4). The arguments will

follow [7, 18, 19], where the case of a single Schr¨ odinger equations has been considered. The

main ingredients are the conservation laws of (1.1) and of the Hamiltonian associated with

the ODE in (1.4) and a modulational stability property for a suitable class of ground state

solutions for the associated autonomous elliptic system (E), recently proved in [28] by the

authors in the same spirit of the works [31, 32] on scalar Schr¨ odinger equations.

The problem for the single equation has been also studied using the WKB analysis (see

for example [9] and the references therein), to our knowledge, there are no results for the

system using this approach. Some of the arguments and estimates in the paper are strongly

based upon those of [19]. On the other hand, for the sake of self-containedness, we prefer

to include all the details in the proofs.

1.2. Admissible ground state solutions. Let Hεbe the space of the vectors Φ = (φ1,φ2)

in H = H1(RN;C2) endowed with the rescaled norm

?Φ?2

Hε=

1

εN?Φ?2

2and ?φi?2

2=?φi(x)¯φi(x)dx.

2+

1

εN−2?∇Φ?2

2= ?φi?2

2,

where ?Φ?2

Lebesgue space L2given by ?φi?2

We aim to study the semiclassical dynamics of a least energy solution of problem (E) once

the action of external forces is taken into consideration.

In [3, 22, 30] it is proved that there exists a least action solution R = (r1,r2) ?= (0,0)

of (E) which has nonnegative components. Moreover, R is a solution to the following

minimization problem (cf. [23, Theorems 3.4 and 3.6])

2= ?(φ1,φ2)?2

2= ?φ1?2

2+ ?φ2?2

L2 is the standard norm in the

(1.5)

E(R) = min

ME,where

M = {U ∈ H : ?U?2= ?R?2},

where the functional E : H → R is defined by

E(U) =1

2?∇U?2

1

p + 1

2−

|u1|2p+2+ |u2|2p+2+ 2β|u1|p+1|u2|p+1?

?

Fβ(U)dx(1.6)

Fβ(U) =

?

, (1.7)

Page 4

4E. MONTEFUSCO, B. PELLACCI, AND M. SQUASSINA

for any U = (u1,u2) ∈ H. We shall denote with G the set of the (complex) ground state

solutions.

Remark 1.1. Any element V = (v1,v2) of G has the form

V (x) = (eiθ1|v1(x)|,eiθ2|v2(x)|),

for some θ1,θ2∈ S1(so that (|v1|,|v2|) is a real, positive, ground state solution). Indeed, if

we consider the minimization problems

x ∈ RN,

σC= inf?E(V ) : V ∈ H, ?V ?L2 = ?R?L2?,

σR= inf?E(V ) : V ∈ H1(RN;R2)?V ?L2 = ?R?L2?

it results that σC= σR. Trivially one has σC≤ σR. Moreover, if V = (v1,v2) ∈ H, due to

the well-known pointwise inequality |∇|vi(x)|| ≤ |∇vi(x)| for a.e. x ∈ RN, it holds

?

so that also E(|v1|,|v2|) ≤ E(V ). In particular, we conclude that σR≤ σC, yielding the de-

sired equality σC= σR. Let now V = (v1,v2) be a solution to σCand assume by contradiction

that, for some i = 1,2,

|∇|vi(x)||2dx ≤

?

|∇vi(x)|2dx,i = 1,2,

LN({x ∈ RN: |∇|vi|(x)| < |∇vi(x)|}) > 0,

where LNis the Lebesgue measure in RN. Then ?(|v1|,|v2|)?L2 = ?V ?L2, and

σR≤1

2

i=1

2

?

?

|∇|vi||2dx −

?

Fβ(|v1|,|v2|)dx <1

2

2

?

i=1

?

|∇vi|2dx −

?

Fβ(v1,v2)dx = σC,

which is a contradiction, being σC = σR. Hence, we have |∇|vi(x)|| = |∇vi(x)| for a.e.

x ∈ RNand any i = 1,2. This is true if and only if Revi∇(Imvi) = Imvi∇(Revi). In

turn, if this last condition holds, we get

¯ vi∇vi= Revi∇(Revi) + Imvi∇(Imvi),

which implies that Re(i¯ vi(x)∇vi(x)) = 0 a.e. in RN. Finally, for any i = 1,2, from this last

identity one immediately finds θi∈ S1with vi= eiθi|vi|, concluding the proof.

In the scalar case, the ground state solution for the equation

a.e. in RN,

(1.8)

−1

2∆r + r = r2p+1

in RN

is always unique (up to translations) and nondegenerate (see e.g. [20, 25, 31]). For sys-

tem (E), in general, the uniqueness and nondegeneracy of ground state solutions is a delicate

open question.

The so called modulational stability property of ground states solutions plays an important

rˆ ole in soliton dynamics on finite time intervals. More precisely, in the scalar case, some

delicate spectral estimates for the seld-adjoint operator E′′(r) were obtained in [31, 32],

allowing to get the following energy convexity result.

Page 5

SOLITON DYNAMICS FOR CNLS SYSTEMS WITH POTENTIALS5

Theorem 1.2. Le r be a ground state solution of equation (1.8) with p < 2/N. Let φ ∈

H1(RN,C) be such that ?φ?2= ?r?2and define the positive number

Γφ= inf

θ∈[0,2π)

y∈RN

?φ(·) − eiθr(· − y))?2

H1.

Then there exist two positive constants A and C such that

Γφ≤ C(E(φ) − E(R)),

provided that E(φ) − E(R) < A.

For systems, we consider the following definition.

Definition 1.3. We say that a ground state solution R = (r1,r2) of system (E) is admissible

for the modulational stability property to hold, and we shall write that R ∈ R, if ri∈ H2(RN)

are radial, |x|ri∈ L2(RN), the corresponding solution φi(t) belongs to H2(RN) for all times

t > 0 and the following property holds: let Φ ∈ H be such that ?Φ?2= ?R?2and define the

positive number

(1.9)ΓΦ:=inf

θ1,θ2∈[0,2π)

y∈RN

?Φ(·) − (eiθ1r1(· − y),eiθ2r2(· − y))?2

H.

Then there exist a continuous function ρ : R+→ R+with

constant C such that

ρ(ξ)

ξ

→ 0 as ξ → 0+and a positive

ρ(ΓΦ) + ΓΦ≤ C(E(Φ) − E(R)).

In particular, there exist two positive constants A and C′such that

(1.10)ΓΦ≤ C′(E(Φ) − E(R)),

provided that ΓΦ< A.

In the one dimensional case, for an important physical class, there exists a ground state

solution of system (E) which belongs to the class R (see [28]).

Theorem 1.4. Assume that N = 1, p ∈ [1,2) and β > 1. Then there exists a ground state

solution R = (r1,r2) of system (E) which belongs to the class R.

1.3. Statement of the main results. The action of external forces is represented by a

potential V : RN→ R satisfying

(1.11)V is a C3function bounded with its derivatives,

and we will study the asymptotic behavior (locally in time) as ε → 0 of the solution of the

following Cauchy problem

iε∂tφ1+ε2

2∆φ1− V (x)φ1+ φ1(|φ1|2p+ β|φ2|p+1|φ1|p−1) = 0

iε∂tφ2+ε2

2∆φ2− V (x)φ2+ φ2(|φ2|2p+ β|φ1|p+1|φ2|p−1) = 0

?x − ˜ x

(Sε)

in RN× R+,

in RN× R+,

φ1(x,0) = r1

ε

?

e

i

εx·˜ξ,φ2(x,0) = r2

?x − ˜ x

ε

?

e

i

εx·˜ξ,

Page 6

6E. MONTEFUSCO, B. PELLACCI, AND M. SQUASSINA

where ˜ x,˜ξ ∈ RNN ≥ 1, the exponent p is such that

(1.12)0 < p < 2/N

It is known (see [15]) that, under these assumptions, and for any initial datum in L2, there

exists a unique solution Φε= (φε

We have chosen as initial data a scaling of a real vector R = (r1,r2) belonging to R.

The first main result is the following

1,φε

2) of the Cauchy problem that exists globally in time.

Theorem 1.5. Let R = (r1,r2) be a ground state solution of (E) which belongs to the class

R. Under assumptions (1.11), (1.12), let Φε= (φε

(Sε). Furthermore, let (x(t),ξ(t)) be the solution of the Hamiltonian system

1,φε

2) be the family of solutions to system

(1.13)

˙ x(t) = ξ(t)

˙ξ(t) = −∇V (x(t))

x(0) = ˜ x

ξ(0) =˜ξ.

Then, there exists a locally uniformly bounded family of functions θε

such that, defining the vector Qε(t) = (qε

i: R+→ S1, i = 1,2,

1(x,t),qε

2(x,t)) by

qε

i(x,t) = ri

?x − x(t)

ε

?

e

i

ε[x·ξ(t)+θε

i(t)],

it holds

(1.14)

?Φε(t) − Qε(t)?Hε≤ O(ε),

as ε → 0

locally uniformly in time.

Roughly speaking, the theorem states that, in the semiclassical regime, the modulus of

the solution Φεis approximated, locally uniformly in time, by the admissible real ground

state (r1,r2) concentrated in x(t), up to a suitable phase rotation. Theorem 1.5 can also be

read as a description of the slow dynamic of the system close to the invariant manifold of

the standing waves generated by ground state solutions. This topic has been studied, for

the single equation, in [29].

Remark 1.6. Suppose that˜ξ = 0 and ˜ x is a critical point of the potential V . Then the

constant function (x(t),ξ(t)) = (˜ x,0), for all t ∈ R+, is the solution to system (1.13). As a

consequence, from Theorem 1.5, the approximated solutions is of the form

ri

?x − ˜ x

ε

?

e

i

εθε

i(t),x ∈ RN, t > 0,

that is, in the semiclassical regime, the solution concentrates around the critical points of

the potential V . This is a remark related to [27] where we have considered as initial data

ground states solutions of an associated nonautonomous elliptic problem.

Page 7

SOLITON DYNAMICS FOR CNLS SYSTEMS WITH POTENTIALS7

Remark 1.7. As a corollary of Theorem 1.5 we point out that, in the particular case of a

constant potential, the approximated solution has components

?x − ˜ x −˜ξt

Hence, the mass center x(t) of Φ(t,x) moves with constant velocity˜ξ realizing a uniform

motion. This topic has been tackled, for the single equation, in [31].

ri

ε

?

e

i

ε[x·˜ξ+θε

i(t)],x ∈ RN, t > 0.

Remark 1.8. For values of β > 1 both components of the ground states R are nontrivial

and, for R ∈ R, the solution of the Cauchy problem are approximated by a vector with

both nontrivial components. We expect that ground state solutions for β > 1 are unique

(up to translations in RN) and nondegenerate.

We can also analyze the behavior of total momentum density defined by

(1.15)Pε(x,t) := pε

1(x,t) + pε

2(x,t),for x ∈ RN, t > 0,

where

(1.16)pε

i(x,t) :=

1

εN−1Im?φ

ε

i(x,t)∇φε

i(x,t)?,for i = 1,2, x ∈ RN, t > 0.

Moreover, let M(t) := (m1+ m2)ξ(t) be the total momentum of the particle x(t) solution

of (1.13), where

(1.17)mi:= ?ri?2

2,for i = 1,2.

The information about the asymptotic behavior of Pεand of the mass densities |φε

are contained in the following result.

i|2/εN

Theorem 1.9. Under the assumptions of Theorem 1.5, there exists ε0> 0 such that

??(|φε

for every ε ∈ (0,ε0) and locally uniformly in time.

Remark 1.10. Essentially, the theorem states that, in the semiclassical regime, the mass

densities of the components φiof the solution Φεbehave as a point particle located in x(t) of

mass respectively miand the total momentum behaves like M(t)δx(t). It should be stressed

that we can obtain the asymptotic behavior for each single mass density, while we can only

afford the same result for the total momentum. The result will follow by a more general

technical statement (Theorem 2.4).

1|2/εNdx,|φε

??Pε(t,x)dx − M(t)δx(t)

2|2/εNdx) −(m1,m2)δx(t)

??

(C2×C2)∗≤ O(ε2),

(C2)∗≤ O(ε2),

??

Remark 1.11. The hypotheses on the potential V can be slightly weakened. Indeed, we

can assume that V is bounded from below and that ∂αV are bounded only for |α| = 2 or

|α| = 3. This allows to include the important class of harmonic potentials (used e.g. in

Bose-Einstein theory), such as

V (x) =1

2

N

?

j=1

ω2

jx2

j,ωj∈ R, j = 1,...,N.

Page 8

8E. MONTEFUSCO, B. PELLACCI, AND M. SQUASSINA

Hence, equation (1.13) reduces to the system of harmonic oscillators

(1.18)¨ xj(t) + ω2

jxj(t) = 0,j = 1,...,N.

For instance, in the 2D case, renaming x1(t) = x(t) and x2(t) = y(t) the ground states

solutions are driven around (and concentrating) along the lines of a Lissajous curves having

periodic or quasi-periodic behavior depending on the case when the ratio ωi/ωjis, respec-

tively, a rational or an irrational number. See Figures 1 and 2 below for the corresponding

phase portrait in some 2D cases, depending on the values of ωi/ωj.

x(t)

y(t)

1.51 0.50 -0.5 -1-1.5

3

2

1

0

-1

-2

-3

x(t)

y(t)

1.510.50 -0.5-1-1.5

3

2

1

0

-1

-2

-3

Figure 1. Phase portrait of system (1.18) in 2D with ω1/ω2= 3/5 (left) and

ω1/ω2= 7/5 (right). Notice the periodic behaviour.

x(t)

y(t)

1.51 0.50-0.5 -1 -1.5

3

2

1

0

-1

-2

-3

x(t)

y(t)

1.510.50 -0.5-1 -1.5

3

2

1

0

-1

-2

-3

Figure 2. Phase portrait of system (1.18) in 2D with ω1/ω2=√3/3 increas-

ing the integration time from t ∈ [0,40π] (left) to t ∈ [0,60π] (right). Notice

the quasi-periodic behaviour, the plane is filling up.

The paper is organized as follows.

In Section 2 we set up the main ingredients for the proofs as well as state two technical

approximation results (Theorems 2.2, 2.4) in a general framework. In Section 3 we will

collect some preliminary technical facts that will be useful to prove the results. In Section 4

we will include the core computations regarding energy and momentum estimates in the

semiclassical regime. Finally, in Section 5, the main results (Theorems 1.5 and 1.9) will be

proved.

Page 9

SOLITON DYNAMICS FOR CNLS SYSTEMS WITH POTENTIALS9

2. A more general Schr¨ odinger system

In the following sections we will study the behavior, for sufficiently small ε, of a solution

Φ = (φ1,φ2) of the more general Schr¨ odinger system

(Fε)

iε∂tφ1+ε2

2∆φ1− V (x)φ1+ φ1(|φ1|2p+ β|φ2|p+1|φ1|p−1) = 0

iε∂tφ2+ε2

2∆φ2− W(x)φ2+ φ2(|φ2|2p+ β|φ1|p+1|φ2|p−1) = 0

?x − ˜ x

where p verifies (1.12), the potentials V, W both satisfy (1.11) and (r1,r2) is a real ground

state solution of problem (E). As for the case of a single potential, we get a unique globally

defined Φε= (φε

1]). Moreover, if the initial data are chosen in H2×H2, then Φε(t) enjoys the same regularity

property for all positive times t > 0 (see e.g. [10]).

in RN× R+,

in RN× R+,

φ1(0,x) = r1

ε

?

e

i

εx·˜ξ1

φ2(0,x) = r2

?x − ˜ x

ε

?

e

i

εx·˜ξ2,

1,φε

2) that depends continuously on the initial data (see, e.g. [15, Theorem

Remark 2.1. With no loss of generality, we can assume V,W ≥ 0. Indeed, if φ1,φ2is a

solution to (Fε), since V,W are bounded from below by (1.11), there exist µ > 0 such that

V (x) + µ ≥ 0 and W(x) + µ ≥ 0, for all x ∈ RN. Thenˆφ1= φ1e−iµt

solution of (Fε) with V + µ (resp. W + µ) in place of V (resp. W).

ε andˆφ2= φ2e−iµt

ε is a

We will show that the dynamics of (φε

1,φε

2) is governed by the solutions

X = (x1,x2) : R → R2N,Ξ = (ξ1,ξ2) : R → R2N,

of the following Hamiltonian systems

(H)

˙ x1(t) = ξ1(t)

˙ξ1(t) = −∇V (x1(t))

(x1(0),ξ1(0)) = (˜ x,˜ξ1),

H2(t) =1

˙ x2(t) = ξ2(t)

˙ξ2(t) = −∇W(x2(t))

(x2(0),ξ2(0)) = (˜ x,˜ξ2).

Notice that the Hamiltonians related to these systems are

(2.1)H1(t) =1

2|ξ1(t)|2+ V (x1(t)),

2|ξ2(t)|2+ W(x2(t))

and are conserved in time. Under assumptions (1.11) it is immediate to check that the

Hamiltonian systems (H) have global solutions. With respect to the asymptotic behavior

of the solution of (Fε) we can prove the following results.

2.1. Two more general results. We now state two technical theorems that will yield, as

a corollary, Theorems 1.5 and 1.9.

Theorem 2.2. Assume (1.12) and that V, W both satisfy (1.11). Let Φε= (φε

the family of solutions to system (Fε). Then, there exist ε0 > 0, Tε

continuous functions ̺ε: R+→ R with ̺ε(0) = O(ε2), locally uniformly bounded sequences

1,φε

2) be

∗> 0, a family of

Page 10

10E. MONTEFUSCO, B. PELLACCI, AND M. SQUASSINA

of functions θε

(qε

i: R+→ S1and a positive constant C, such that, defining the vector Qε(t) =

2(x,t)) by

1(x,t), qε

qε

i(x,t) = ri

?x − x1(t)

ε

?

e

i

ε[x·ξi(t)+θε

i(t)],i = 1,2

it results

?Φε(t) − Qε(t)?Hε≤ C

?

̺ε(t) +

?̺ε(t)

ε

?2

,

for all ε ∈ (0,ε0) and all t ∈ [0,Tε

system for V in (H).

∗], where x1(t) is the first component of the Hamiltonian

Remark 2.3. Theorem 2.2 is quite instrumental in the context of our paper, as we cannot

guarantee in the general case of different potentials that the function ̺εis small as ε vanishes,

locally uniformly in time. Moreover, the time dependent shifting of the components qiinto

x1(t) is quite arbitrary, a similar statement could be written with the component x2(t) in

place of x1(t), this arbitrariness is a consequence of the same initial data ˜ x in (H) for both

x1and x2. The task of different initial data in (H) for x1and x2is to our knowledge an

open problem.

In the following, if ξiare the second components of the systems in (H), we set

(2.2)M(t) := m1ξ1(t) + m2ξ2(t), t > 0.

If Φε= (φε

1,φε

2) is the family of solutions to (Fε), we have the following

Theorem 2.4. There exist ε0> 0 and Tε

R+→ R with ̺ε(0) = O(ε2) such that

??(|φε

∗> 0 and a family of continuous functions ̺ε:

1|2/εNdx,|φε

2|2/εNdx) −(m1,m2)δx1(t)

??

(C2×C2)∗≤ ̺ε(t),

??Pε(t,x)dx − M(t)δx1(t))??

(C2)∗≤ ̺ε(t),

for every ε ∈ (0,ε0) and all t ∈ [0,Tε

∗].

3. Some preliminary results

In this section we recall and show some results we will use in proving Theorems 1.5, 1.9,

2.2 and 2.4. First we recall the following conservation laws.

Proposition 3.1. The mass components of a solution Φ of (Fε),

(3.1)

Nε

i(t) :=

1

εN?φε

i(t)?2

L2,

for i = 1,2, t > 0,

are conserved in time. Moreover, also the total energy defined by

(3.2)Eε(t) = Eε

1(t) + Eε

2(t)

Page 11

SOLITON DYNAMICS FOR CNLS SYSTEMS WITH POTENTIALS11

is conserved as time varies, where

Eε

1(t) =

1

2εN−2?∇φε

1

2εN−2?∇φε

1?2

L2 +

1

εN

1

εN

?

V (x)|φε

1|2dx −

1

2εN

1

2εN

?

?

Fβ(Φε)dx,

Eε

2(t) =

2?2

2+

?

W(x)|φε

2|2dx −

Fβ(Φε)dx.

Proof. This is a standard fact. For the proof, see e.g. [15].

Remark 3.2. From the preceding proposition we obtain that, due to the form of our initial

data, the mass components Nε

Nε

εN

i(t) do not actually depend on ε. Indeed, for i = 1,2,

1

?

i/εN/2have constant norm in L2equal, respectively, to mi. In The-

orem 2.4 we will show that, for sufficiently small values of ε, the mass densities behave,

point-wise with respect to t, as a δ functional concentrated in x1(t).

(3.3)

i(t) = Nε

i(0) =

|φε

i(x,0)|2dx =

1

εN

????ri

?x − ˜ x

ε

????

2dx = mi.

Thus, the quantities φε

In the following we will often make use of the following simple Lemma.

Lemma 3.3. Let A ∈ C2(RN) be such that A,DjA,D2

R = (r1,r2) be a ground state solution of problem E. Then, for every y ∈ RNfixed, there

exists a positive constant C0such that

ijA are uniformly bounded and let

(3.4)

????

?

[A(εx + y) − A(y)]r2

i(x)dx

????≤ C0ε2.

Proof. By virtue of the regularity properties of the function A and Taylor expansion The-

orem we get

1

ε2

????

?

[A(εx + y) − A(y)]r2

i(x)dx

????≤1

+ ?Hes(A)?∞

ε|∇A(y)|

????

?

?

xr2

i(x)dx

????

|x|2r2

i(x)dx

where ?Hes(A)?∞denotes the L∞norm of the Hessian matrix associated to the function

A. The first integral on the right hand side is zero since each component riis radial. The

second integral is finite, since |x|ri∈ L2(RN).

In order to show the desired asymptotic behavior we will use the following property of

the functional δyon the space C2(RN).

Lemma 3.4. There exist K0, K1, K2positive constants, such that, if ?δy−δz?C2∗≤ K0then

K1|y − z| ≤ ?δy− δz?C2∗ ≤ K2|y − z|

Proof. For the proof see [19, Lemma 3.1, 3.2].

The following lemma will be used in proving our main result.

Page 12

12E. MONTEFUSCO, B. PELLACCI, AND M. SQUASSINA

Lemma 3.5. Let Φε= (φε

αi: R → RNdefined by

1,φε

2) be a solution of (Fε) and consider the vector functions

(3.5)αε

i(t) =

?

pε

i(x,t)dx − miξi(t),t > 0, i = 1,2,

where the ξis are defined in (H) and the mis are defined in (1.17), for i = 1,2. Then

{t ?→ αε

Remark 3.6. The integral in (3.5) defines a vector whose components are the integral of

Im(φε

Proof. The continuity of αiimmediately follows from the regularity properties of the solu-

tion φε

i(t)} is a continuous function and αε

i(0) = 0, for i = 1,2.

i∂φε

i/∂xj)/εN−1for j = 1,...,N, so that αε

i: R → RN.

i. In order to complete the proof, first note that, for all x ∈ RN,

¯φε

ε

so that, as riis a real function, the conclusion follows by a change of variable.

i(x,0)∇φε

i(x,0) =i

˜ξir2

i

?x − ˜ x

ε

?

+1

εri

?x − ˜ x

ε

?

∇ri

?x − ˜ x

ε

?

,

Lemma 3.7. Let V and W both satisfying assumptions (1.11) and let Φε= (φε

solution of (Fε). Moreover, let A a positive constant defined by

1,φε

2) be a

(3.6)A = K1sup

[0,T0][|x1(t)| + |x2(t)|] + K0

where xi(t) is defined in (H), K0and K1are defined in Lemma 3.4, and let χ be a C∞(RN)

function such that 0 ≤ χ ≤ 1 and

(3.7)χ(x) = 1

if |x| < A,

Then the functions

χ(x) = 0

if |x| > 2A.

(3.8)

ηε

1(t) = m1V (x1(t)) −

1

εN

?

?

χ(x)V (x)|φε

1(x,t)|2dx,

ηε

2(t) = m2W(x2(t)) −

1

εN

χ(x)W(x)|φε

2(x,t)|2dx.

are continuous and satisfy |ηε

Proof. The continuity of ηε

tion φε

an analogous way. We have

i(0)| = O(ε2) for i = 1,2.

iimmediately follows from the regularity properties of the solu-

i. We will prove the conclusion only for ηε

1(0), the result for ηε

2(0) can be showed in

|ηε

1(0)| =

????m1V (x1(0)) −

1

εN

?

?

χ(x)V (x)|φε

?x − ˜ x

?x − ˜ x

1(x,0)|2dx

????

dx.

????

≤

????m1V (˜ x) −

εN

|x|>A

1

εN

V (x)r2

1

ε

?

dx

+

1

?

(1 − χ(x))V (x)r2

1

ε

?

Page 13

SOLITON DYNAMICS FOR CNLS SYSTEMS WITH POTENTIALS 13

Then, by Lemma 3.3, and a change of variables imply

|ηε

1(0)| ≤ O(ε2) +

?

(1 − χ(˜ x + εy))V (˜ x + εy)r2

1(y)dy.

The properties of χ and r1and assumption (1.11) yield the conclusion.

We will also use the following identities.

Lemma 3.8. The following identities holds for i = 1,2.

(3.9)

1

εN

∂|φε

∂t

i|2

(x,t) = −divxpε

i(x,t),x ∈ RN, t > 0.

Moreover, for all t > 0, it results

(3.10)

?

∂Pε

∂t(x,t)dx = −1

εN

?

∇V (x)|φε

1(x,t)|2dx −

1

εN

?

∇W(x)|φε

2(x,t)|2dx,

where Pε(x,t) is the total momentum density defined in (1.15).

Remark 3.9. It follows from identity (3.10) that for systems with constant potentials the

total momentum?Pεdx is a constant of motion.

Remark 3.10. As evident from identity (3.10) as well as physically reasonable, in the case

of systems of Schr¨ odinger equations, the balance for the momentum needs to be stated for

the sum Pεinstead on the single components pε

proof, where the coupling terms appear.

i. See also identities (3.11) and (3.12) in the

Proof. In order to prove identity (3.9) note that

−divxpε

i= −

1

εN−1Im(¯φε

i∆φε

i),

1

εN

∂|φε

∂t

i|2

=

2

εNRe((φε

i)t¯φε

i)

Since φε

by¯φε

of the nonlinearity.

ε1−NIm(φ

isolves the corresponding equation in system (Fε), we can multiply the equation

iand add this identity to its conjugate; the conclusion follows from the properties

Concerning identity (3.10), observe first that, setting (pε

ε

1(x,t)∂jφε

1)j(x,t) =

1(x,t)) for any j and ∂j= ∂xj, it holds

∂(pε

∂t

1)j

= ε1−NIm(∂tφ

= ε1−NIm(∂tφ

= 2ε1−NIm(∂tφ

ε

1∂jφε

1) + ε1−NIm(φ

1∂jφε

ε

1∂j(∂tφε

1))

ε

1) + ε1−NIm(∂j

1∂jφε

?φ

?φ

ε

1∂tφε

1

?) − ε1−NIm(∂jφ

1

?).

ε

1∂tφε

1)

ε

1) + ε1−NIm(∂j

ε

1∂tφε

Page 14

14E. MONTEFUSCO, B. PELLACCI, AND M. SQUASSINA

In particular the second term integrates to zero. Concerning the first addendum, take the

first equation of system (Fε), conjugate it and multiply it by 2ε−N∂jφ1. It follows

ε∂jφε

2ε1−NIm(∂tφ1

1) = −ε2−NRe(∆φ1

− 2ε−N|φε

ε∂jφε

1) + 2ε−NV (x)Re(φ1

ε∂jφε

ε∂jφε

1)

1|2pRe(φ1

1) − 2βε−N|φε

2|p+1|φε

?|∂iφε

1|2

?|φε

1|p−1Re(φ1

1|2

?

ε∂jφε

1)

= −ε2−NRe(∂i

+ ε−N∂j

?∂iφ1

1|2?− ε−N∂jV (x)|φε

p + 1

ε∂jφε

1)) + ε2−N∂j

2

?V (x)|φε

?|φε

− ε−N∂j

1|2p+2

?

− 2βε−N|φε

2|p+1∂j

1|p+1

p + 1

?

.

Of course, one can argue in a similar fashion for the second component φ2. Then, taking

into account that all the terms in the previous identity but ∂jV (x)|φε

integrate to zero due to the H2regularity of φ1, we reach

?

?

Adding these identities for any j and taking into account that by the regularity properties

of φε

1|2and |φε

2|p+1∂j|φε

1|p+1

∂(pε

∂t

∂(pε

∂t

1)j

dx = −1

dx = −1

εN

?

?

∂V

∂xj(x)|φε

∂W

∂xj(x)|φε

1|2dx −2β

εN

?

?

|φε

2|p+1∂j

?|φε

?|φε

1|p+1

p + 1

?

?

dx(3.11)

2)j

εN

2|2dx −2β

εN

|φε

1|p+1∂j

2|p+1

p + 1

dx.(3.12)

iit holds?∂j(|φε

1|p+1|φε

2|p+1)dx = 0, formula (3.10) immediately follows.

4. Energy, mass and momentum estimates

4.1. Energy estimates in the semiclassical regime. In order to obtain the desired

asymptotic behavior stated in Theorems 1.5, 1.9, 2.2 and 2.4, we will first prove a key

inequality concerning the functional E defined in (1.6). As pointed out in the introduction,

the main ingredients involved are the conservations laws of the Schr¨ odinger system and

of the Hamiltonians functions and a modulational stability property for admissible ground

states.

The idea is to evaluate the functional E on the vector Υε= (vε

are given by

1,vε

2) whose components

(4.1)vε

i(x,t) = e−i

εξi(t)·[εx+x1(t)]φε

i(εx + x1(t),t)

where X = (x1,x2), Ξ = (ξ1,ξ2) are the solution of the system (H). More precisely, we will

prove the following result.

Theorem 4.1. Let Φε= (φε

defined in (4.1). Then, there exist ε0and Tε

t ∈ [0,Tε

(4.2)

1,φε

2) be a family of solutions of (Fε), and let Υεbe the vector

∗such that for every ε ∈ (0,ε0) and for every

∗), it holds

0 ≤ E(Υε) − E(R) ≤ αε+ ηε+ O(ε2),

where we have set

(4.3)αε(t) =??(ξ1(t),ξ2(t)) · (αε

1(t),αε

2(t))??,ηε(t) = |ηε

1(t) + ηε

2(t)|,

Page 15

SOLITON DYNAMICS FOR CNLS SYSTEMS WITH POTENTIALS15

αi, ηiare given in (3.5), (3.8) and R = (r1,r2) is the real ground state belonging to the class

R taken as initial datum in (Fε). Moreover, there exist families of functions θε

positive constant L such that

?x − yε

for every ε ∈ (0,ε0) and all t ∈ [0,Tε

Proof. By a change of variable and Proposition 3.1, we get

i, yε

1and a

(4.4)

???Φε−

?

e

i

ε(xξ1+θε

1)r1

1

ε

?

∗).

, e

i

ε(xξ2+θε

2)r2

?x − yε

1

ε

?????

2

Hε≤ L?αε+ ηε+ O(ε2)?,

(4.5)

?vε

i(·,t)?2

2= ?φε

i(εx + x1(t),t)?2

2=

1

εN?φε

i(·,t)?2

2= mi,t > 0, i = 1,2,

where miare defined in (1.17). Hence the mass of vε

Moreover, by a change of variable, and recalling definition (1.16) we have

1

2εN−2?∇Φε?2

?

Then, taking into account the form of the total energy functional, we obtain

1

εN

?

Moreover, using Proposition 3.1 and performing a change of variable we get

iis conserved during the evolution.

E(Υε) =

2+1

2

?m1|ξ1|2+ m2|ξ2|2?−

1(x,t),pε

1

εNFβ(Φε)

−

(ξ1(t),ξ2(t)) · (pε

2(x,t))dx.

E(Υε) = Eε(t) −

??V (x)|φε

1|2+ W(x)|φε

2|2?dx +1

2

?m1|ξ1|2+ m2|ξ2|2?

−

(ξ1(t),ξ2(t)) · (pε

1(x,t),pε

2(x,t))dx.

Eε(t) = Eε(0) = Eε?

= E(R) +1

??V (εx + ˜ x)|r1|2+ W(εx + ˜ x)|r2|2?dx,

this joint with Lemma 3.3 and the conservation of the Hamiltonians Hi(t) yield

E(Υε) − E(R) =1

2

?

r1

?x − ˜ x

ε

?

e

i

εx·˜ξ1,r2

?x − ˜ x

ε

?

e

i

εx·˜ξ2?

2

?m1|˜ξ1|2+ m2|˜ξ2|2?

+

?

m1(|˜ξ1(t)|2+ |ξ1(t)|2) + m2(|˜ξ2(t)|2+ |ξ2(t)|2)

?

−

(ξ1(t),ξ2(t)) · (pε

1(x,t),pε

2(x,t))dx

+ m1V (˜ x) + m2W(˜ x) −

?|ξ1(t)|2+ V (x1(t))?+ m2

−

1

εN

1

εN

??V (x)|φε

?|ξ2(t)|2+ W(x2(t))?

1(x,t),pε

1|2+ W(x)|φε

2|2?dx

=m1

?

(ξ1(t),ξ2(t)) · (pε

2(x,t))dx

−

??V (x)|φε

1|2+ W(x)|φε

2|2?dx + O(ε2)