Variational properties and orbital stability of standing waves for NLS equation on a star graph
We study standing waves for a nonlinear Schrödinger equation on a star graph i.e. half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength . The nonlinearity is of focusing power type. The dynamics is given by an equation of the form , where is the Hamiltonian operator which generates the linear Schrödinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every . Furthermore, we determine the ground states, as minimizers of the action on the Nehari manifold, and order the various families. Finally, we show that the ground states are orbitally stable for every allowed if the nonlinearity is subcritical or critical, and for otherwise.
Keywords: quantum graphs, non-linear Schrödinger equation, solitary waves.
MSC 2010: 35Q55, 81Q35, 37K40, 37K45.
In the present paper a rigorous analysis of the stationary behavior of nonlinear Schrödinger equation (NLS) on a graph is given, beginning from the simplest type of unbounded graph, the star graph. In a previous paper [ACFN1] the authors studied the behavior in time of an asymptotically solitary solution of NLS resident on a single edge of the graph in the far past, and impinging on the vertex with various types of couplings, giving a quantitative analysis of reflection and transmission of the solitary wave after the collision at the junction. Here we concentrate on a different phenomenon, namely the existence of persistent nonlinear bound states on the graph (localized, or pinned nonlinear modes), and on their orbital stability, when an attractive interaction is present at the vertex. Some of the results here discussed and proved were briefly announced in [ACFN2] .
Let us briefly give a collocation of the model in the physical context. Generally speaking, one can consider the NLS as a paradigm for the behavior of nonlinear dispersive equations, but it is also an ubiquitous model appearing in several concrete physical situations. The main fields of application which we have in mind are the propagation of electromagnetic pulses in nonlinear media (typically laser beams in Kerr media or signal propagation in optical fibers), and dynamics of Bose-Einstein condensates (BEC). We are interested in the way solutions of NLS are affected by the presence of inhomogeneities of various type. The propagation on the line in the presence of defects has been a subject of intense study in the last years and it gives rise to quite interesting phenomena, such as defect induced modes [CM, FMO, ANS], i.e. standing solutions strongly localized around the defect. The presence of defect modes affects propagation by allowing trapping of wave packets, as experimentally shown in the case of local photonic potentials in [Linzon]. On the other hand, nonlinearity can induce escaping of solitons from confining potentials, as demonstrated in [Pe]. A last interesting phenomenon is the strong alteration of tunneling through potential barriers in the presence of nonlinear defocusing optical media [Wan]. In this paper we consider NLS propagation through junctions in networks. For example, when the dynamics of a BEC takes place in essentially one-dimensional substrates (“cigar shaped” condensates) or a laser pulse propagates in optical fibers and thin waveguides, the question arises of the effect of a ramified junction on propagation and on the possible generation of stable bound states. The analysis of the behavior of NLS on networks is not yet a fully developed subject, but it is currently growing. Concerning situations of direct physical interest we mention the analysis of scattering at Y junctions (“beam splitters”) and other network configurations (“ring interferometers”) for one dimensional Bose liquids discussed in [TOD]. Some more results are known for the discrete chain NLS model (DNLS), see in connection with the present paper the analysis in [Miro]. Other recent developments are in [GSD, Sob]. In particular, in the paper [GSD] scattering from a complex network sustaining nonlinear Schrödinger dynamics is studied in relation to characterization of quantum chaos.
With these phenomenological and analytical premises in mind we would like to construct a mathematical model capable to represent, in a schematic but rigorous way, the propagation and stationary behavior of a nonlinear Schrödinger field at a junction of a network. We begin by giving the needed preliminaries to rigorously define our model. We recall that the linear Schrödinger equation on graphs has been for a long time a very developed subject due to its applications in quantum chemistry, nanotechnologies and more generally mesoscopic physics. Standard references are [[BCFK06], [BEH], [Kuc04], [Kuc05], [KS99], EKKST08, BK13], where more extensive treatments are given. Here we recall only the definitions needed to have a self-contained exposition. We consider a graph constituted by infinite half-lines attached to a common vertex. The natural Hilbert space where to pose a Schrödinger dynamics is then . Elements in will be represented as function vectors with components in , namely
We denote the elements of by capital Greek letters, while functions in are denoted by lowercase Greek letters. We say that is symmetric if does not depends on . The norm of -functions on is naturally defined by
From now on for the -norm on the graph we drop the subscript and simply write . Accordingly, we denote by the scalar product in .
Analogously, given , we define the space as the set of functions on the graph whose components are elements of the space , and the norm is correspondingly defined by
Besides, we need to introduce the spaces
equipped with the norms
Whenever a functional norm refers to a function defined on the graph, we omit the symbol .
When an element of evolves in time, we use in notation the subscript : for instance, . Sometimes we shall write in order to highlight the dependence on time, or whenever such a notation is more understandable.
The dynamics we want to set on the graph is generated by a linear part and a nonlinear one. We begin by describing the linear part.
Fixed , we consider a Hamiltonian operator, denoted by and called graph or vertex, defined on the domain
where denotes the derivative of the function with respect to the space variable related to the -th edge. The action of the operator is given by
The Hamiltonian is a selfadjoint operator on ([[KS99]]) and generalizes to the graph the ordinary Schrödinger operator with potential of strength on the line [aghh:05]. Similarly to that case, the interaction is encoded in the boundary condition. The case in (1.2) plays a distinguished role and it defines what is usually given the name of free or Kirchhoff boundary condition; we will indicate the corresponding operator as . Notice that for a graph with two edges, i.e. the line, continuity of wavefunction and its derivative for an element of makes the interaction disappear; this fact justifies the name of free Hamiltonian. A vertex with can be interpreted as the presence of a deep attractive potential well or attractive defect. This interpretation can be enforced by showing that, as in the case of the line, the operator is a norm resolvent limit for vanishing of a scaled Hamiltonian , where and is a positive normalized potential on the graph (see [[BEH]] and reference therein). The attractive character shows in the fact that for every a (single) bound state exists for the linear dynamics, with energy . On the contrary, on a Kirchhoff vertex no bound states exist, the spectrum is purely absolutely continuous, but a zero energy resonance appears. Finally we recall that in the case of repulsive delta interaction , which is however of minor interest here, there are not bound states nor zero energy resonances.
The quadratic form associated to is defined on the finite energy space
and is given by
The corresponding bilinear form is denoted by and explicitly given by
As a particular case, the quadratic form associated to is defined on the same space, that is , and reads
Now let us introduce the nonlinearity. To this end we define where acts “componentwise” as for a suitable and .
We are interested in the special but important case of a power nonlinearity of focusing type, so we choose
After this preparation it is well defined the NLS equation on the graph,
where . This abstract nonlinear Schrödinger equation amounts to a system of scalar NLS equations on the halfline, coupled through the boundary condition at the origin included in the domain (1.2).
In Section 2 we show that for well-posedness of the dynamics described by equation (1.3) (in weak form) for initial data in the finite energy space holds true. Moreover, if then the solution exists for all times and blow-up does not occur. Finally, as in the standard NLS on the line, mass and energy are conserved, where
and analogously, in the case , for the Kirchhoff energy
After setting the model and its well-posedness (see Section 2), we turn to the main subject of this paper, existence and properties of standing wave solutions to (1.3). Standing waves are solutions of the form
The function is the amplitude or the profile (with some abuse of interpretation) of the standing wave, and we will frequently refer to the set of as to the stationary states of the problem.
The amplitude satisfies the stationary equation
This equation has a variational structure.
Let us define the action functional
The Euler-Lagrange equation of the action is the stationary equation above. The action , defined on the form domain of the operator , is unbounded from below. Nevertheless, it is bounded on the so called natural (or Nehari) constraint s.t. , where . Note that and thus the Nehari manifold is a codimension one constraint which contains all the solutions to the stationary equation. One of our main results is the following theorem.
Theorem 1 (Existence of minimizers for the Action functional).
Let . There exists such that for the action functional constrained to the Nehari manifold admits an absolute minimum, i.e. a such that and .
So the action admits a constrained minimum on the natural constraint for every if the strength of the interaction at the vertex is negative and sufficiently strong. The strategy of the proof, which is a consequence of results of Section 3 and Section 4, makes use of non trivial elements and we give here some remarks. To get the existence of the minimum one has at a certain point to compare the action with with the Kirchhoff action .
In Section 3 we prove that the Kirchhoff action, while bounded from below on its natural constraint, has no minimum (see [ACFN3] for an analogous phenomenon affecting the constrained energy functional). As a matter of fact, the infimum can be exactly computed and it is achieved as the limit over a sequence of functions which escape at infinity on a single edge. A main step in establishing the previous picture and in applying it to the case, is the exact calculation of the infimum and the identification of the minimizers of the free action; to this end one exploits an extension and generalization of the classical properties of symmetric rearrangements of and functions to the case of graphs.
In Section 4 we prove Theorem 1. The analysis follows in part proofs of similar results for singular interactions on the line given in [FMO, [AN12]], with major modifications due to the fact that in this case the comparison with the free case is not standard. In particular the upper bound in given in the statement of Theorem 1 is a consequence of the fact that one needs the condition to guarantee the existence of an absolute minimum in the constrained action, penalizing situations analogous to escaping minimizers of the free action. A sufficiently strong attractive interaction at the vertex allows to satisfy the previous condition.
We conjecture that the action has a local constrained minimum that is larger than the infimum when the condition on fails, but presently we do not have a proof of this fact.
In Section 5 an explicit construction of all the stationary states of the problem is obtained, by solving the stationary equation for every value of . It turns out that for every and there exist families of stationary states of different action and energy, which can be ordered in to form a nonlinear spectrum (the family is unique only in the case , i.e. the line). The state of minimal action is the ground state , which is of course the solution to the constrained minimum problem for the action just discussed. The others are excited states, and they exists, for every , when . See Theorem 4 for a complete description.
Finally, in Section 6 we study the stability of ground states. Stability is an important requisite of a standing wave, because at a physical level unstable states are rapidly dominated by dispersion, drift or blow-up and so are undetectable (instability of NLS with a potential on the line is studied, partly numerically, in [LeCFFKS]). The concept of stability, due to gauge or invariance of the action, is orbital stability. The solutions remain close to the orbit of the ground state for all times if they start close enough to it. The framework in which we study orbital stability of the ground state is the mainstream of Weinstein and Grillakis-Shatah-Strauss theory, which applies to infinite dimensional Hamiltonian systems such as abstract NLS equation when a regular branch of standing waves (not necessarily ground states) exists, which is our case.
According to this theory, to guarantee orbital stability one needs to verify a set of spectral conditions on the linearization of the NLS around the ground state, and a slope (or Vakhitov-Kolokolov in the physical literature) condition concerning the behavior in of the -norm of the ground state. Some adaptation of standard methods is needed to treat the singular character of the interaction at the vertex, but in fact it turns out that in the range of over which Theorem is valid, the spectral conditions and the slope condition are encountered for every and every nonlinearity in .
Theorem 2 (Orbital stability of the ground state).
Let , , . Then the ground state is orbitally stable in
The proof of this result is contained in Section 6. Notice that one has orbital stability of the ground state in a range of nonlinearities which includes the critical case. An analogous phenomenon occurs in the case of the line, previously treated in [LeCFFKS]. This marks a difference with the case of a free () NLS on the line, where one has orbital instability in the critical case. Finally, the proof of the previous theorem (see Remark 6.1) shows that for supercritical nonlinearities the ground state is orbitally stable for not too large : there exists a threshold such that one has orbital stability for the ground state with and orbital instability in the opposite case.
Appendix A contains a theory of symmetric rearrangements on star graphs. More precisely, the classical inequalities stating conservation of norms and domination of kinetic energy are proved. This last property, i.e. the Pólya-Szegő inequality, is particularly interesting because it changes with respect to the case of the line through the presence of a factor which takes into account the number of edges of the graph, and this fact is crucial in the previously described analysis of action minimization on a star graph. A previous analysis of rearrangements on bounded graphs is contained in [[Fri05]], and a comparison of the two treatments is given at the end of Appendix A. We stress the fact that the theory of rearrangements is a general tool and it is in principle applicable to more general or different problems.
We end this introduction with a few open problems and future directions of study. Concerning technical issues, a different strategy from the one here pursued in the analysis of ground states and their stability is minimization of energy at constant mass (see the classical paper [[CL]] and for models related to the present one [ANV]); it requires a non trivial extension to graphs of concentration-compactness method and it is studied in [[ACFN4]]. Nothing is known up to now about stability properties of the branches of excited states , which exist for every and sufficiently high ; this is a subject of special interest because there are only few cases where excited states of NLS equations are explicitly known. The authors plan to study this issue in a subsequent paper. Finally it would be interesting, and perhaps a difficult task, the extension of the analysis here given to different classes of graphs, possibly with non trivial topology. Several results in this direction were recently obtained in [AST14-1, AST14-2, CFN14]. Dispersion properties, relevant to give precise large time behavior of solutions have been studied for trees, including star graphs, in [BI1, BI2]. This is a first step for the analysis of possible asymptotic stability of standing waves on networks. All these issues will need the development of new technical tools, both concerning variational analysis and stability properties.
2. Well-posedness of the model
For our purposes it is sufficient to prove that the solution of the Schrödinger equation is uniquely defined in time in the energy domain and that energy and mass are conserved quantities. This section is devoted to the proof of these conservation laws and of the well-posedness of equation (1.3). In fact along the proofs we shall always work with the weak form of (1.3), namely
We consider the problem of the well-posedness in the sense of, e.g., [[Caz]], i.e., we prove existence and uniqueness of the solution to equation (2.1) in the energy domain of the system. Such a domain turns out to coincide with the form domain of the linear part of equation (1.3). We follow the traditional line of proving first local well-posedness, and then extending it to all times by means of a priori estimates provided by the conservation laws. Proceeding as in [ACFN1] where the cubic NLS is treated, we show the well-posedness of the dynamics for any , i.e. local existence and uniqueness for initial data in the energy space. Moreover, we will prove that if , then the well-posedness is global, i.e. the solution exists for all times and no collapse occurs. For a more extended treatment of the analogous problem for a two-edge vertex (namely, the real line with a point interaction at the origin) see [[AN09]].
We endow the energy domain with the -norm defined in (1.1). Moreover we denote by the dual of , i.e. the set of the continuous linear functionals on . We denote the dual product of and by . In such a bracket we sometimes exchange the place of the factor in with the place of the factor in : indeed, the duality product follows the same algebraic rules of the standard scalar product.
As usual, one can extend the action of to the space , with values in , by
where denotes the bilinear form associated to the selfadjoint operator .
Furthermore, for any the identity
holds in too. To prove it, one can first test the functional on an element in the operator domain , obtaining
Then, the result can be extended to by a density argument.
In order to prove a well-posedness result we need to generalize standard one-dimensional Gagliardo-Nirenberg estimates to graphs, i.e.
where the is a positive constant which depends on the index only. The proof of (2.3) follows immediately from the analogous estimates for functions of the real line, considering that any function in can be extended to an even function in , and applying this reasoning to each component of (see also [MPF91, I.31]).
Proposition 2.1 (Local well-posedness in ).
We define the space endowed with the norm Given , we define the map as
We first notice that the nonlinearity preserves the space . Then by and using Hölder and Gagliardo-Nirenberg inequalities, one obtains
Analogously, given , one has
We point out that the constant appearing in (2.4) and (2.5) is independent of , , and . Now let us restrict the map to elements such that . From (2.4) and (2.5), if is chosen to be strictly less than , then is a contraction of the ball in of radius , and so, by the contraction lemma, there exists a unique solution to (2.1) in the time interval . By a standard one-step bootstrap argument one immediately has that the solution actually belongs to , and due to the validity of (1.3) in the space we immediately have that the solution actually belongs to .
The proof of the existence of a maximal solution is standard, while the blow-up alternative is a consequence of the fact that, whenever the -norm of the solution is finite, it is possible to extend it for a further time by the same contraction argument. ∎
The next step consists in the proof of the conservation laws.
Proposition 2.2 (Conservation laws).
Let . For any solution to the problem (2.1), the following conservation laws hold at any time :
The conservation of the -norm can be immediately obtained by the validity of equation (1.3) in the space :
by the selfadjointness of . In order to prove the conservation of the energy, first we notice that is differentiable as a function of time. Indeed,
and then, passing to the limit ,
where we used the selfadjointness of and (1.3). Furthermore,
and the proposition is proved. ∎
Corollary 2.1 (Global well-posedness).
Let . For any , the equation (2.1) has a unique solution .
By estimate (2.3) with and conservation of the -norm, there exists a constant , that depends on only, such that
Therefore a uniform (in ) bound on is obtained. As a consequence, one has that no blow-up in finite time can occur, and therefore, by the blow-up alternative, the solution is global in time. ∎
3. Variational Analysis: the Kirchhoff vertex
In this section we compute the infimum of the action functional for the Kirchhoff case. As often in this framework, the action functional is unbounded from below and we have to restrict it to the Nehari manifold, or natural constraint manifold, in order to have a functional bounded from below. The knowledge of the infimum of the constrained action will be a key ingredient in the next section in the proof of the main theorem.
The strategy of the computation of the infimum is standard: first we derive a lower bound and then we show that this lower bound is optimal by means of a minimizing sequence. In the derivation of the lower bound symmetric rearrangements are used. Using this technique we can map the initial variational problem into a variational problem with symmetric functions which can be reduced to a problem on the halfline providing the required estimate.
The minimizing sequence shows in fact that the constrained action exhibits a sort of spontaneous symmetry breaking in the Kirchhoff case. That is, although the functional is symmetric, the minimizing sequence is localized on a single edge.
As defined in the introduction, in the Kirchhoff case the action functional is given by
while the Nehari functional reads
The Nehari manifold is defined by . The action restricted to the Nehari manifold will be named reduced action and is given by
It is understood that the domain of all the functionals is always .
Theorem 3 (Infimum of the Action for the Kirchhoff case).
The infimum of the action functional restricted to the Nehari manifold is given by:
The proof of (3.2) is divided into two parts: first we derive a lower bound for , then we prove that the lower bound is optimal by means of a minimizing sequence.
In order to derive a lower bound, we consider an auxiliary variational problem with symmetric functions. This is done by using the rearrangements on the graph which are discussed in Appendix LABEL:rearra.
Let and let be its symmetric rearrangement. We known that is positive, symmetric and . Moreover, by Theorem LABEL:polya and Proposition LABEL:lp, we have
Therefore for such that we have
Taking into account (3.1), and the above properties of one can enlarge the domain in the following way in order to lower the infimum,
Under the scaling, , , the last variational problem scales as
It is convenient to choose as
in order to reconstruct a Nehari manifold with a rescaled as constraint. Moreover due to the symmetry of we have
It is convenient to introduce a variational problem on the half line and an auxiliary variational problem on the line. Let and be defined in the following way:
Notice that the following inequality holds true:
Indeed, by absurd, assume and let be a minimizing sequence for the problem on the halfline. We can extend by parity and obtain a sequence such that
Passing to the limit one would obtain
which contradicts our absurd hypothesis. Therefore provides a lower bound for the variational problem we are interested in. On the other hand the exact expression of can be easily obtained from known results (see [[Caz]] Ch. VIII) , and it is given by:
Estimate (3) closes the first part of the proof. Now it is sufficient to exhibit a sequence of trial functions satisfying the constraint and such that . We consider a sequence of soliton-like functions escaping to infinity, i.e.
where is defined in Appendix LABEL:appB by (LABEL:soliton) and is a function such that , for and for . The sequence belongs to but does not satisfy the constraint . It is straightforward to check that
where the r.h.s. of (3.9) depends on and . In the remaining part of the proof we shall not make explicit the dependence on and of the constant appearing in estimates. Let be defined by
It is straightforward to check that . Then in order to prove (3.2), it is sufficient to prove that
Now we prove that
and by (3.9), it is sufficient to prove that
Taking into account (3.8) and (LABEL:soliteq) and integrating by parts one has
The remainder can be estimated using the exponential decay of and in the following way
The last equality follows by dominated convergence and (LABEL:formula2):