Soliton dynamics for CNLS systems with potentials
The semiclassical limit of a weakly coupled nonlinear focusing Schrödinger system in presence of a nonconstant potential is studied. The initial data is of the form with , where is a real ground state solution, belonging to a suitable class, of an associated autonomous elliptic system. For sufficiently small, the solution will been shown to have, locally in time, the form , where is the solution of the Hamiltonian system , with and .
Key words and phrases:Weakly coupled nonlinear Schrödinger systems, concentration phenomena, semiclassical limit, orbital stability of ground states, soliton dynamics
2000 Mathematics Subject Classification:34B18, 34G20, 35Q55
1. Introduction and main result
In recent years much interest has been devoted to the study of systems of weakly coupled nonlinear Schrödinger equations. This interest is motivated by many physical experiments especially in nonlinear optics and in the theory of Bose-Einstein condensates (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 Schrödinger system
with , and is a constant modeling the birefringence effect of the material. The potential is a regular function in modeling the action of external forces (see (1.9)), are complex valued functions and is a small parameter playing the rôle of Planck’s constant. The task to be tackled with respect to this system is to recover the full dynamics of a solution as a point particle subjected to galileian motion for the parameter 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 Schrödinger equation and for a special class of solutions, namely standing wave solutions (see  and the 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 (that is Kerr nonlinearities)
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 has been studied in , where some sufficient and necessary conditions have been established. To our knowledge the semiclassical dynamics of different kinds of solutions of a single Schrödinger equation has been tackled in the series of papers [7, 18, 19] (see also  for recent developments on the long term soliton dynamics), assuming that the initial datum is of the form , where is the unique ground state solution of an associated elliptic problem (see equation (1.6)) and . 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 –up to translations and phase changes–and the translations and phase changes are precisely related with the solution of a Newtonian system in governed by the gradient of the potential . Here we want to recover similar results for system (1.1) taking as initial data
where the vector is a suitable ground state (see Definition 1.3) of the associated elliptic system
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 ) for real ground state solutions of (1.1) holds. This is expected, at least in the case where . 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 (1.1) (for a suitable ground state ) 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 converge–in the dual space of –to the delta measure with mass given by and concentrated along , solution to the (driving) Newtonian differential equation
where and 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 (1.2). In the case of constant our statements are related with the results obtained, by linearization procedure, in  for the single equation. Here, by a different approach, we show that (1.2) gives a modulation equation for the solution generated by the initial data (1.1). Although we cannot predict the shape of the solution, we know that the dynamic of the mass center is described by (1.2). The arguments will follow [7, 18, 19], where the case of a single Schrödinger equations has been considered. The main ingredients are the conservation laws of (1.1) and of the Hamiltonian associated with the ODE in (1.2) and a modulational stability property for a suitable class of ground state solutions for the associated autonomous elliptic system (1.1), recently proved in  by the authors in the same spirit of the works [31, 32] on scalar Schrödinger equations.
The problem for the single equation has been also studied using the WKB analysis (see for example  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 . 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 be the space of the vectors in endowed with the rescaled norm
and is the standard norm in the Lebesgue space given by .
We aim to study the semiclassical dynamics of a least energy solution of problem once the action of external forces is taken into consideration.
In [3, 22, 30] it is proved that there exists a least action solution of (1.1) which has nonnegative components. Moreover, is a solution to the following minimization problem (cf. [23, Theorems 3.4 and 3.6])
where the functional is defined by
for any . We shall denote with the set of the (complex) ground state solutions.
Any element of has the form
for some (so that is a real, positive, ground state solution). Indeed, if we consider the minimization problems
it results that . Trivially one has . Moreover, if , due to the well-known pointwise inequality for a.e. , it holds
so that also . In particular, we conclude that , yielding the desired equality . Let now be a solution to and assume by contradiction that, for some ,
where is the Lebesgue measure in . Then , and
which is a contradiction, being . Hence, we have for a.e. and any . This is true if and only if . In turn, if this last condition holds, we get
which implies that a.e. in . Finally, for any , from this last identity one immediately finds with , concluding the proof.
In the scalar case, the ground state solution for the equation
The so called modulational stability property of ground states solutions plays an important rôle in soliton dynamics on finite time intervals. More precisely, in the scalar case, some delicate spectral estimates for the seld-adjoint operator were obtained in [31, 32], allowing to get the following energy convexity result.
Le be a ground state solution of equation (1.6) with . Let be such that and define the positive number
Then there exist two positive constants and such that
provided that .
For systems, we consider the following definition.
We say that a ground state solution of system (1.1) is admissible for the modulational stability property to hold, and we shall write that , if are radial, , the corresponding solution belongs to for all times and the following property holds: let be such that and define the positive number
Then there exist a continuous function with as and a positive constant such that
In particular, there exist two positive constants and such that
provided that .
Assume that , and . Then there exists a ground state solution of system (1.1) which belongs to the class .
1.3. Statement of the main results
The action of external forces is represented by a potential satisfying
and we will study the asymptotic behavior (locally in time) as of the solution of the following Cauchy problem
where , the exponent is such that
It is known (see ) that, under these assumptions, and for any initial datum in , there exists a unique solution of the Cauchy problem that exists globally in time. We have chosen as initial data a scaling of a real vector belonging to .
The first main result is the following
Let be a ground state solution of (1.1) which belongs to the class . Under assumptions (1.9), (1.10), let be the family of solutions to system (1.3). Furthermore, let be the solution of the Hamiltonian system
Then, there exists a locally uniformly bounded family of functions , , such that, defining the vector by
locally uniformly in time.
Roughly speaking, the theorem states that, in the semiclassical regime, the modulus of the solution is approximated, locally uniformly in time, by the admissible real ground state concentrated in , 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 .
Suppose that and is a critical point of the potential . Then the constant function , for all , is the solution to system (1.11). As a consequence, from Theorem 1.5, the approximated solutions is of the form
that is, in the semiclassical regime, the solution concentrates around the critical points of the potential . This is a remark related to  where we have considered as initial data ground states solutions of an associated nonautonomous elliptic problem.
For values of both components of the ground states are nontrivial and, for , the solution of the Cauchy problem are approximated by a vector with both nontrivial components. We expect that ground state solutions for are unique (up to translations in ) and nondegenerate.
We can also analyze the behavior of total momentum density defined by
Moreover, let be the total momentum of the particle solution of (1.11), where
The information about the asymptotic behavior of and of the mass densities are contained in the following result.
Under the assumptions of Theorem 1.5, there exists such that
for every and locally uniformly in time.
Essentially, the theorem states that, in the semiclassical regime, the mass densities of the components of the solution behave as a point particle located in of mass respectively and the total momentum behaves like . 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).
The hypotheses on the potential can be slightly weakened. Indeed, we can assume that is bounded from below and that are bounded only for or . This allows to include the important class of harmonic potentials (used e.g. in Bose-Einstein theory), such as
Hence, equation (1.11) reduces to the system of harmonic oscillators
For instance, in the 2D case, renaming and 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 is, respectively, 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 .
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.
2. A more general Schrödinger system
In the following sections we will study the behavior, for sufficiently small , of a solution of the more general Schrödinger system
where verifies (1.10), the potentials both satisfy (1.9) and is a real ground state solution of problem (1.1). As for the case of a single potential, we get a unique globally defined that depends continuously on the initial data (see, e.g. [15, Theorem 1]). Moreover, if the initial data are chosen in , then enjoys the same regularity property for all positive times (see e.g. ).
We will show that the dynamics of is governed by the solutions
of the following Hamiltonian systems
Notice that the Hamiltonians related to these systems are
and are conserved in time. Under assumptions (1.9) it is immediate to check that the Hamiltonian systems () have global solutions. With respect to the asymptotic behavior of the solution of (2) we can prove the following results.
2.1. Two more general results
Assume (1.10) and that both satisfy (1.9). Let be the family of solutions to system (2). Then, there exist , , a family of continuous functions with , locally uniformly bounded sequences of functions and a positive constant , such that, defining the vector by
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 is small as vanishes, locally uniformly in time. Moreover, the time dependent shifting of the components into is quite arbitrary, a similar statement could be written with the component in place of , this arbitrariness is a consequence of the same initial data in () for both and . The task of different initial data in () for and is to our knowledge an open problem.
If is the family of solutions to (2), we have the following
There exist and and a family of continuous functions with such that
for every and all .
3. Some preliminary results
The mass components of a solution of (2),
are conserved in time. Moreover, also the total energy defined by
is conserved as time varies, where
Proof. This is a standard fact. For the proof, see e.g. .
From the preceding proposition we obtain that, due to the form of our initial data, the mass components do not actually depend on . Indeed, for ,
Thus, the quantities have constant norm in equal, respectively, to . In Theorem 2.4 we will show that, for sufficiently small values of , the mass densities behave, point-wise with respect to , as a functional concentrated in .
In the following we will often make use of the following simple Lemma.
Let be such that are uniformly bounded and let be a ground state solution of problem 1.1. Then, for every fixed, there exists a positive constant such that
Proof. By virtue of the regularity properties of the function and Taylor expansion Theorem we get
where denotes the norm of the Hessian matrix associated to the function . The first integral on the right hand side is zero since each component is radial. The second integral is finite, since .
In order to show the desired asymptotic behavior we will use the following property of the functional on the space .
There exist positive constants, such that, if then
Proof. For the proof see [19, Lemma 3.1, 3.2].
The following lemma will be used in proving our main result.
Let be a solution of (2) and consider the vector functions defined by