Soliton dynamics for CNLS systems with potentials
ABSTRACT The soliton dynamics in the semiclassical limit for a weakly coupled nonlinear focusing Schr\"odinger systems in presence of a nonconstant potential is studied by taking as initial data some rescaled ground state solutions of an associate elliptic system. Comment: 25 pages, 2 figures

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ABSTRACT: We study the spectral structure of the complex linearized operator for a class of nonlinear Schr\"odinger systems, obtaining as byproduct some interesting properties of nondegenerate ground state of the associated elliptic system, such as being isolated and orbitally stable. Comment: 18 pages, 1 figureCommunications on Pure and Applied Analysis 05/2009; · 0.59 Impact Factor  SourceAvailable from: Benedetta Pellacci[Show abstract] [Hide abstract]
ABSTRACT: Orbital stability property for weakly coupled nonlinear Schr\"odinger equations is investigated. Different families of orbitally stable standing waves solutions will be found, generated by different classes of solutions of the associated elliptic problem. In particular, orbitally stable standing waves can be generated by least action solutions, but also by solutions with one trivial component whether or not they are ground states. Moreover, standing waves with components propagating with the same frequencies are orbitally stable if generated by vector solutions of a suitable single Schr\"odinger weakly coupled system, even if they are not ground states. Comment: 21 pages, original articleAdvanced Nonlinear Studies 09/2008; · 0.54 Impact Factor  SourceAvailable from: Raffaella Servadei[Show abstract] [Hide abstract]
ABSTRACT: The soliton dynamics for a general class of nonlinear focusing Schrödinger problems in presence of nonconstant external (local and nonlocal) potentials is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation.Journal of Mathematical Analysis and Applications 01/2010; 365:776796. · 1.05 Impact Factor
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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 BoseEinstein 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(φ12p+ βφ2p+1φ1p−1) = 0
iε∂tφ2+ε2
2∆φ2− V (x)φ2+ φ2(φ22p+ βφ1p+1φ2p−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(r12p+ βr2p+1r1p−1)
2∆r2+ r2= r2(r22p+ βr1p+1r2p−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 selfcontainedness, 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−
u12p+2+ u22p+2+ 2βu1p+1u2p+1?
?
Fβ(U)dx(1.6)
Fβ(U) =
?
, (1.7)
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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) = (eiθ1v1(x),eiθ2v2(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 wellknown 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
?
?
∇vi2dx −
?
Fβ(v1,v2)dx <1
2
2
?
i=1
?
∇vi2dx −
?
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= eiθivi, 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 seldadjoint operator E′′(r) were obtained in [31, 32],
allowing to get the following energy convexity result.
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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, xri∈ 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(φ12p+ βφ2p+1φ1p−1) = 0
iε∂tφ2+ε2
2∆φ2− V (x)φ2+ φ2(φ22p+ βφ1p+1φ2p−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.
i2/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).
12/εNdx,φε
??Pε(t,x)dx − M(t)δx(t)
22/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
BoseEinstein 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 quasiperiodic 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 11.5
3
2
1
0
1
2
3
x(t)
y(t)
1.510.50 0.511.5
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 0.500.5 1 1.5
3
2
1
0
1
2
3
x(t)
y(t)
1.510.50 0.51 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 quasiperiodic 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(φ12p+ βφ2p+1φ1p−1) = 0
iε∂tφ2+ε2
2∆φ2− W(x)φ2+ φ2(φ22p+ βφ1p+1φ2p−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
2ξ1(t)2+ V (x1(t)),
2ξ2(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:
12/εNdx,φε
22/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
12dx −
1
2εN
1
2εN
?
?
Fβ(Φε)dx,
Eε
2(t) =
2?2
2+
?
W(x)φε
22dx −
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,
pointwise 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
????
x2r2
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 xri∈ 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
K1y − z ≤ ?δy− δz?C2∗ ≤ K2y − 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
i2
(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
i2
=
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−Nφε
ε∂jφε
1) + 2ε−NV (x)Re(φ1
ε∂jφε
ε∂jφε
1)
12pRe(φ1
1) − 2βε−Nφε
2p+1φε
?∂iφε
12
?φε
1p−1Re(φ1
12
?
ε∂jφε
1)
= −ε2−NRe(∂i
+ ε−N∂j
?∂iφ1
12?− ε−N∂jV (x)φε
p + 1
ε∂jφε
1)) + ε2−N∂j
2
?V (x)φε
?φε
− ε−N∂j
12p+2
?
− 2βε−Nφε
2p+1∂j
1p+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
12and φε
2p+1∂jφε
1p+1
∂(pε
∂t
∂(pε
∂t
1)j
dx = −1
dx = −1
εN
?
?
∂V
∂xj(x)φε
∂W
∂xj(x)φε
12dx −2β
εN
?
?
φε
2p+1∂j
?φε
?φε
1p+1
p + 1
?
?
dx(3.11)
2)j
εN
22dx −2β
εN
φε
1p+1∂j
2p+1
p + 1
dx.(3.12)
iit holds?∂j(φε
1p+1φε
2p+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
?m1ξ12+ m2ξ22?−
1(x,t),pε
1
εNFβ(Φε)
−
(ξ1(t),ξ2(t)) · (pε
2(x,t))dx.
E(Υε) = Eε(t) −
??V (x)φε
12+ W(x)φε
22?dx +1
2
?m1ξ12+ m2ξ22?
−
(ξ1(t),ξ2(t)) · (pε
1(x,t),pε
2(x,t))dx.
Eε(t) = Eε(0) = Eε?
= E(R) +1
??V (εx + ˜ x)r12+ W(εx + ˜ x)r22?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˜ξ12+ m2˜ξ22?
+
?
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
12+ W(x)φε
22?dx
=m1
?
(ξ1(t),ξ2(t)) · (pε
2(x,t))dx
−
??V (x)φε
12+ W(x)φε
22?dx + O(ε2)
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