Hylomorphic solitons on lattices

This paper is devoted to prove the existence of solitons on lattices. We are interested in solitary waves and solitons whose existence is related to the ratio energy/charge. These solitary waves are called hylomorphic. This class includes the Q-balls, which are spherically symmetric solutions of the nonlinear Klein-Gordon equation, as well as solitary waves and vortices which occur, by the same mechanism, in the nonlinear Schroedinger equation and in gauge theories. In this paper we prove an abstract existence theorem which applies to many situations already considered in the literature and also to the nonlinear Schroedinger (and Klein-Gordon) equations defined on a lattice.


Introduction
Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behavior (see e.g. [1], [3], [8], [15], [19], [20]).
Following [4], [2], [3], the third type of solitary waves or solitons will be called hylomorphic. This class includes the Q-balls which are spherically symmetric solutions of NKG and vortices which might be defined as spinning Q-balls. Also it includes solitary waves and vortices which occur, by the same mechanism, in NS and in gauge theories; a bibliography on this subject can be found in the review papers [8], [3], and, for the vortices, in [9]. This paper is devoted to the proof of a general abstract theorem which can be applied to the main situations considered in the literature (see e.g. [3] and [8]). However this theorem can be also applied to the NS and to the NKG defined on a lattice (namely eq. (72) when V satisfies (78) and eq. (90) when W satisfies (93)). These results are new.
The paper is organized as follows. In section 2 we give the definition of hylomorphic solitons and describe their general features; in section 3 we prove some abstract results on the existence of hylomorphic solitons; in section 4 and in section 5 we apply the abstract results to NS and to NKG defined on a lattice.

Hylomorphic solitary waves and solitons 2.1 An abstract definition of solitary waves and solitons
Solitary waves and solitons are particular orbits of a dynamical system described by one or more partial differential equations. The states of this system are described by one or more fields which mathematically are represented by functions where V is a vector space with norm | · | V which is called the internal parameters space. We denote our dynamical system by (X, T ) where X is the set of the states and T : R × X → X is the time evolution map. If u 0 ∈ X, the evolution of the system will be described by the function Now we can give a formal definition of solitary wave: Definition 1 An orbit U (t, x) is called solitary wave if it has the following form: where γ (t) : R N → R N is a one parameter group of isometries which depends smoothly on t and is a group of (linear) transformation of the internal parameter space which depends smoothly on t and x. In particular, if γ (t) x = x, U is called standing wave.
For example, consider a solution of a field equation which has the following form In this case In this paper we are interested in standing waves, so (3) takes the form The solitons are solitary waves characterized by some form of stability. To define them at this level of abstractness, we need to recall some well known notions from the theory of dynamical systems.
Definition 2 Let (X, d) be a metric space and let (X, T ) be a dynamical system.
Now we are ready to give the definition of (standing) soliton: where K ⊂ X is a compact set and F ⊆ R N is not necessarily compact.
A generic field u ∈ Γ can be written as follows where θ belongs to a set of indices Ξ which parametrize K and x 0 ∈ F. In the generic case of many concrete situations, Γ is a manifold, then, (iii) implies that it is finite dimensional and (θ, x 0 ) are a system of coordinates.
For example, in the case (3), we have The proof of (ii) of definition 3, namely that a solitary wave has enough stability to be a soliton, is a delicate question and in many cases of interest it is open. Moreover the notion of stability depends on the choice of the space X and on the choice of its metric d and hence different choices might lead to more or less satisfactory answers.

Integrals of motion and hylomorphic solitons
The existence and the properties of hylomorphic solitons are guaranteed by the interplay between energy E and another integral of motion which, in the general case, is called hylenic charge and it will be denoted by C. Notice that E and C can be considered as functionals defined on the phase space X.
Thus, we make the following assumptions on the dynamical system (X, T ) : • A-1. It is variational, namely the equations of the motions are the Euler-Lagrange equations relative to a Lagrangian density L[u].
• A-2. The equations are invariant for time translations.
• A-3. The equations are invariant for a S 1 -action acting on the internal parameter space V (cf. (1)).
By A-1, A-2 and A-3 and Noether's theorem (see e.g. [14], [3], [8]) it follows that our dynamical system has 2 first integrals: • the invariance with respect to time translations gives the conservation of the energy which we will shall denote by E(u); • The invariance A-3 gives another integral of motion called hylenic charge which we shall denote by C(u).

Now we set
Using the definition 3, we get the definition of hylomorphic soliton as follows: Definition 4 Let U be a soliton according to definition 3. U is called a (standing) hylomorphic soliton if Γ (as defined in (5)) coincides with the set of minima of E on M σ , namely Remark 5 Suppose that the Lagrangian L[u] is invariant for (a representation of ) the Lorentz or the Galileo group. Then given a standing hylomorphic soliton, we can get a hylomorphic travelling soliton just by Galileo or a Lorentz transformation respectively.
We recall that in physics literature the solitons of definition 4 are called Qballs [13] and were first studied in the pioneering paper [16]. The existence of stable solitary waves in particular cases has been extablished in [12] and [17]. The existence of hylomorphic solitons in more general equations has been proved in [2].
If the energy E is unbounded from below on M σ it is still possible ( [18], [11]) to have standing wave (see def. 1). Moreover there are also cases [17] in which it is possible to have solitons (see def. 3) which are only local minimizers [10]. These solitons are not hylomorphic (def. 4) and they can be destroyed by a perturbation which send them out of the basin of attraction.
In the next section we analyze some abstract situations which imply Γ σ = ∅ and the existence of hylomorphic solitons (definition 4).

The general framework
We assume that E and C are two functionals on D R N , V (≡ C ∞ 0 (R N )) defined by densities. This means that, given u ∈ D R N , V , there exist density functions ρ E,u (x) and ρ C,u (x) ∈ L 1 (R N ) i. e. functions such that Also we assume that the energy can be written as follows E,u is quadratic in u and ρ E,u contains the higher order terms. If we assume ρ (2) E,u > 0 for u = 0, then we can define the following norm: and the Hilbert space We assume that the energy E and the charge C can be extended as functional of class C 2 in X; in particular we will write E as follows: where L : X → X ′ is the duality operator, namely Lu, u = u 2 and K is superquadratic. Also, we assume that where L 0 is a linear operator and K 0 is superquadratic. For any Ω ⊂ R N we will write In the general scheme described above, we make the following assumptions: • (E-0) (the main protagonists) E and C are two functionals on X of the form (7) and (8) • (E-1) (lattice translation invariance) the charge and the energy are lattice translation invariant.
where T z : X → X is a linear representation of the additive group Z N defined as follows: A is an invertible matrix which characterizes the representation T z . Such a T z will be called lattice transformation.

The main theorems
In the framework of the previous section, we want to investigate sufficient conditions which guarantee that the energy has a minimum on the set In this section and in the next one we will study this minimization problem, namely we may think of E and C as two abstract functionals. In section 3.4 we will apply the minimization result to the case in which E and C are just the energy and the hylenic charge of a dynamical system.
We set ., N and (10) where A is the matrix in (9). Also we set The value e 0 = +∞ is allowed. We now set, Theorem 6 Assume (E-0,..,E-4). Moreover assume that Then, there existsσ such that where Γσ is as in definition 4. Moreover • if u n is a sequence such that Λ (u n ) → Λσ and |C (u n )| →σ then • Γσ has the structure in (5), namely with K compact and F ⊂ R N is a closed set such that F = j +F, ∀j ∈ Z N .
• any u ∈ Γσ solves the eigenvalue problem In many concrete situations C(u) and Λ (u) behave monotonically with respect to the action of the dilatation group R θ defined by In this case we obtain a stronger result: Theorem 7 Let the assumptions of Th. 6 be satisfied. Moreover suppose that there is the action of a group R θ , θ ∈ R + such that Λ (R θ u) is decreasing in θ while E(R θ u) and C(R θ u) are increasing. Then there exists σ 0 such that, for anyσ ≥ σ 0 , the same conclusions of Th. 6 hold.
The following proposition gives an expression of e 0 (see (12)) which will be useful in the applications of Theorems 6 and 7.

Proposition 8
Let E and C be as in (7) and (8); then Proof. We have

Proof of the main theorems
We shall first prove some technical lemmas.
Lemma 9 Let Q be defined by (11) and T j q = q + Aj (q ∈ Q). Then Proof. Take a generic x ∈ R N and set y = A −1 x. We can decompose y as follows: y = q 0 + j where j ∈ Z N and q 0 ∈ Q 0 defined by (10). Then where q := Aq 0 ∈ Q. Since x is generic the lemma is proved.
where Λ σ is defined in (14), is lower semicontinuous. Moreover Proof. The semicontinuity of Λ σ is an immediate consequence of the definition. Moreover, by its definition, we have that (23) follows. Let us prove (24). We set Let u n ∈ X be a sequence such that We can assume, passing eventually to a subsequence, that If such a subsequence does not exist, we have C(u n ) ≤ 0 and we argue in a similar way. Now take ε > 0, then, for n large enough where Now, for every n large, it is possible to take j n ∈ I such that To show this, we argue indirectly and assume that then you get This contradicts (26).Now set v n (x) = u n (x + Aj n ).
Then (27) gives Since u n and consequently also u n Q are infinitesimal, from (30) and the definition of e 0 we obtain that and so e 0 ≤ e * Now set L = lim σ→0 Λ σ ; then there exists a sequence u n such that |C (u n )| → 0 if E (u n ) → 0, we have that u n → 0. So, by the definition of e * and (31), we have that L ≥ e * ≥ e 0 .
Lemma 11 (Splitting property) Let E and C be as in (7) and (8) and assume that E-3, E-4 are satisfied. Let w n ⇀ 0 weakly and let u ∈ X; then Proof. We have to show that lim Let us consider each piece independently: Choose ε > 0 and R = R(ε) > 0 such that Then, by the local compactness assumption E-3 (see section 3.1), we have that By (E-4) and the intermediate value theorem we have that for a suitable θ ∈ (0, 1) and since ε is arbitrary, this limit is 0. Then we have proved the splitting property for E. The splitting property for C is obtained arguing in the same way we did with K.
• Any u ∈ Γσ solves the eigenvalue problem Proof . By (32) Let u n ∈ X be a sequence such that In order to fix the ideas, we may assume that If, on the contrary no subsequence of C (u n ) converge toσ, then C (u n ) → −σ and we argue in a similar way. The proof consists of two steps.
Step1. We prove that for a suitable sequence {z n } ⊂ Z N we have whereū = 0 and w n (x) ⇀ 0 weakly in X.
We decompose R N as in (22). Take ε > 0, then, for n large enough Arguing as in (26) and (27) (replacing e * with Λσ ), for n large, it is possible to take j n ∈ I such that E Ωj n (u n ) C Ωj n (u n ) ≤ Λσ + ε.
We set v n (x) = u n (x + Aj n ).
By (40) and (41), E(u n ) and C(u n ) are bounded. So, also E(v n ) and C(v n ) are bounded and, by (E-2), v n is bounded. Letū be the weak limit of v n . We want to show thatū = 0.
Clearly C Q (v n ) = C Ωj n (u n ) . Then, since j n ∈ I, we have that C Q (v n ) > 0 and, for n large, by (44) we have We claim that the sequence C Q (v n ) does not converge to 0; in fact if we have that v n Q → 0; so, by definition of e 0 , and by (46), we have and this fact contradicts (39) if ε > 0 is small enough. Since C Q (v n ) does not converge to 0, by (E-3) with Ω = Q, we have that C Q (ū) > 0 and we can conclude thatū = 0. Now set w n = v n −ū and so w n (x) ⇀ 0 weakly in X.
Step 2. Next we will prove that v n →ū strongly in X namely that w n → 0 strongly in X. So, by (7), it will be enough to show that By (40), (41) and lemma 11 and so Now we set We consider three cases.
Finally (37) clearly follows by the definition of Γσ .
Proof of Th. 6. We prove that the assumptions (32) and (33) of Lemma 12 are satisfied.

Remark 13
By the proof of this theorem, we can see that the assumption Λ * > 0 is used only to get (58). This assumption can be replaced by the following one In fact (64) implies (58). To show this, we argue indirectly and assume that there exists a sequence σ n → ∞ such that σ n Λ σ n is bounded; so there exists a sequence u n such that and By (65) and (E-4), we have that (for a subsequence) u n → ∞; then, by (64), E (u n ) → ∞. This contradicts (66), then we conclude that (58) holds.

Dynamical consequences of the main theorem
The above theorems can be applied to the case in which (X, · ) is the state space of a dynamical system (X, T ) and it proves the existence of hylomorphic solitons; more exactly we have: Theorem 14 Let (X, T ) be a dynamical system and let E and C be the energy and the charge. If X, E and C are as in section 3.1 and satisfy the assumptions of theorem 6, then (X, T ) has hylomorphic solitons. Moreover, if also the assumptions of Th. 7 are satisfied, there exists σ 0 such that there are solitons for any chargeσ ≥ σ 0 .
Proof. We consider Def. 4. We set By theorem 6 Γ σ = ∅. In order to prove the existence of solitons we need to prove (ii) and (iii) of definition 3. (ii) follows by (36). In order to prove stability, we use the Lyapunov criterium; we define the Lyapunov function V : X → R as follows Then, by the Lyapunov stability theorem Γ is stable. The second statements follows directly from Th. 7.

The nonlinear Schrödinger equation
We are interested to the nonlinear Schrödinger equation: where ψ : .

Existence results
We assume that W has the following form where h 2 = W ′′ (0) and N (s) = o(s 2 ). We make the following assumptions on W : such that for any s > 0 : If we set then the hylomorphy assumption (W-iii) reads This assumption implies that ∃s : N (s) < 0.
We make the following assumptions on V : for all x ∈ R N and z ∈ Z N .
Here we want to use the results of the previous sections to study (72). In this case the state u coincides with ψ and the general framework of the previous sections takes the following form: where H 1 (R N ) is the usual Sobolev space and Then the energy E and the hylenic charge C have the form (7) and (8) respectively. We shall prove the following theorem Theorem 15 Assume that W satisfies W-i),...W-iiii) and that V satisfies V-i), V-ii). Moreover assume that where α and h have been introduced in (76) and (75). Then equation (72) admits hylomorphic solitons (see definition 4).

Remark 16
Observe that, when V = 0, assumption (82) reduces to the request α < h, which is the "usual" hylomorphy condition (see [2], [4], [7], [3]). Moreover, in this case it is possible to apply Th. 7 and to get the existence of solitons for any sufficiently large charge.
Remark 17 Actually, the assumptions (W-i,...,W-iiii) are not the most general. For example the positivity assumption is not necessary. In the case V = 0, we refer to [5]. If V = 0, we do not know whether the assumptions used in [5] are sufficient.
We first obtain some estimates on e 0 and Λ * defined by (21) and (15).
Proof of Theorem 15: By (83), (84) and (82) we deduce that 0 < Λ * < e 0 . It can be shown, by standard calculations (see e.g. [7]), that under the assumptions W-i),...,W-iiii) and V-i), V-ii), the functionals E and C, defined by (79) and (81), satisfy (E-0,..,E-4) of section 3.1. Then, by using Theorem 6, we deduce that equation (72) admits hylomorphic solitons. Since these solitons u 0 are minimizers of the energy E on the manifold u ∈ H 1 (R N ) : C(u) = u 2 dx = σ , we get where ω is a Lagrange multiplier. Then it can be easily seen that u 0 solves (72) and u 0 = ψ 0 (x)e −iωt , where ω ∈ R and ψ 0 (x) solve the equation The nonlinear Klein-Gordon equation In this section we will apply th. 6 to the existence of hylomorphic solitons of the nonlinear Klein-Gordon equation. We point out that the existence of such solitons for this equation has been recently stated in [2]. Here we consider the case in which W depends on x and it has a lattice symmetry. More exactly, we consider the equation and W ′ is the derivative with respect to the second variable as in (73). We can write W as follows and N (x, s) = o(s 2 ). We make the following assumptions on W : for all x ∈ R N and z ∈ Z N .
• (NKG-iii) (Hylomorphy) ∃α,s ∈ R + such that W (x,s) ≤ 1 2 α 2s2 • (NKG-iiii)(Growth condition) there are constants c 1 , c 2 > 0, 2 < p < 2N/(N − 2) such that for any s > 0 : We shall assume that the initial value problem is well posed for (NKG). Eq. (90) is the Euler-Lagrange equation of the action functional The energy and the charge take the following form: (the sign "minus"in front of the integral is a useful convention).

The NKG as a dynamical system
We set and we will denote the generic element of X by u = (ψ (x) ,ψ (x)); then, by the well posedness assumption, for every u ∈ X, there is a unique solution ψ(t, x) of (90) such that Thus, using our notation, we can write x)) ∈ C 1 (R, X).
Using this notation, we can write equation (90) in Hamiltonian form: The energy and the charge, as functionals defined in X, become We shall tacitely assume that W is such that E, C are C 1 in X.
In order to prove the existence of hylomorphic solitons, we will use Th. 6. Clearly the energy E and the hylenic charge C have the form (7) and (8) respectively, with L 0 u, u = C(u) = − Re iψψ dx; K 0 (u) = 0.