Hylomorphic solitons in the nonlinear Klein-Gordon equation
Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localised packet and which preserves this localisation in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behaviour. In this paper we show a new mechanism which might produce solitary waves and solitons for a large class of equations, such as the nonlinear Klein-Gordon equation. We show that the existence of these kind of solitons, that we have called hylomorphic solitons, depends on a suitable energy/charge ratio. We show a variational method that allows to prove the existence of hylomorphic solitons and that turns out to be very useful for numerical applications. Moreover we introduce some classes of nonlinearities which admit hylomorphic solitons of different shapes and with different relations between charge, energy and frequency.
- 1 Introduction
- 2 The abstract theory
- 3 The Nonlinear Klein-Gordon equation
- 4 Q-balls
- 5 Acknowledgments
Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localised packet and which preserves this localisation in time.
A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behaviour.
Today, we know (at least) three mechanism which might produce solitary waves and solitons:
Complete integrability; e.g. Korteweg-de Vries equation
Topological constraints: e.g. Sine-Gordon equation
Ratio energy/charge: e.g. the following nonlinear Klein-Gordon equation
This paper is devoted to the third type of solitons which will be called hylomorphic solitons. This class of solitons that are characterised by a suitable energy/charge ratio includes the -balls, spherically symmetric solutions of NKG, as well as solitary waves which occur in the nonlinear Schrödinger equation (see e.g. , ) and in gauge theories (see e.g. , , ). We have chosen the name hylomorphic, which comes from the Greek words “hyle”=“matter”=“set of particles” and “morphe”=“form”, with the meaning of solitons “giving a suitable form to condensed matter” (see Section 2.2 for definitions and details).
The aims of this paper are the following:
to give the definition of hylomorphic solitons and to set this notion in the literature on non-topological solitons;
to describe a new variational approach that allows to prove the existence of hylomorphic solitons for a large class of nonlinearities for NKG. This variational method turns out to be very useful for numerical simulations;
to classify the nonlinearities which give hylomorphic solitons: this classification is based on the different shapes of the solitons and the different relations between charge, energy and frequency. We obtain four classes of nonlinearities and we prove necessary conditions for the nonlinear term to belong to a given class. Numerical simulations show the different behaviour of these classes quantitatively.
2 The abstract theory
2.1 An abstract definition of soliton
We consider dynamical systems with phase space described by one or more fields, which mathematically are represented by a function
where is a vector space with norm which is called the internal parameters space. We assume the system to be deterministic, and denote by the time evolution map, which is assumed to be defined for all . The dynamical system is denoted by . If is the initial condition, the evolution of the system is described by
We assume that namely for every . Moreover for states which satisfy , it is possible to give the notion of barycenter of the state as follows
The term solitary wave is usually used for solutions of field equations whose energy is localised and the localisation of the energy packet is preserved under the evolution. Using the notion of barycenter, we give a formal definition of solitary wave.
A state is called solitary wave if for any there exists a radius such that for all
where denotes the ball in of radius and centre in .
The solitons are solitary waves characterised by some form of stability. To define them at this level of abstractness, we need to recall some well known notions in the theory of dynamical systems.
Let be a metric space and let be a dynamical system. An invariant set is called stable, if for any there exists such that if then for all .
A state is called orbitally stable if there exists a finite dimensional manifold with , such that is invariant and stable for the dynamical system .
The above definition needs some explanation. Since is invariant, is in for all . Thus, since is finite dimensional, the evolution of is described by a finite number of parameters Thus the dynamical system behaves as a point in a finite dimensional phase space. By the stability of , the evolution of a small perturbation of might become very different from , but remains close to . Thus, the perturbed system appears as a finite dimensional system with a small perturbation which depends on an infinite number of parameters.
A state , is called soliton if it is a orbitally stable solitary wave.
According to this definition a soliton is a state in which the mass is “concentrated” around the barycenter . In general, and hence, the “state” of a soliton is described by parameters which define its position and other parameters which define its “internal states”.
2.2 Definition of hylomorphic solitons
In this section we will expose a general method to prove the existence of non-topological solitons. This method leads in a natural way to the definition of hylomorphic solitons.
We make the following assumptions: (i) there are at least two integrals of motion, the energy and the hylomorphic charge ; (ii) the system is invariant for space translations. In some models the hylomorphic charge is just the “usual” charge and this fact justifies this name. We add the attribute hylomorphic just to recall that might not be a charge (as in the case of the nonlinear Schrödinger equation). Moreover, in many models in quantum field theory, represents the expected number of particles, hence for charged particles it is proportional to the electric charge.
Now, given the set
In order to prove the existence of solitary waves and solitons, we follow a method based on the following steps:
(S-1) prove that the energy has a minimum on ;
(S-2) prove that the barycenter (see (2.1)) is well defined and that for any there exists such that
where is the set of minimisers;
(S-3) prove that is finite dimensional;
(S-4) prove that is stable.
These steps will be explained in details for the nonlinear Klein-Gordon equation in Section 4.2.
The integrals of motion are used in the definition of hylomorphic solitons. Setting
and assuming that , we introduce the hylomorphy ratio of a state , defined as
The hylomorphy ratio of a state turns out to be a dimensionless invariant of the motion and an important quantity for the characterisation of solitons.
A soliton is called hylomorphic if
In the following we will refer to (2.5) as the hylomorphy condition. In the study of hylomorphic solitons, the hylomorphy ratio plays a very important role and it is related to a density function which we call binding energy density.
We assume and to be local quantities, namely, given there exist density functions and such that
Then we introduce the binding energy density defined as
The support of
is called the condensed matter region since in these points the binding forces prevail, and the quantity
will be called the condensed matter. The next proposition justifies the choice of the name “hylomorphic”, since it shows that hylomorphic solitons “contain condensed matter”.
If then for all the support of is not empty.
Proof. It follows by the simple relations
If is a finite energy field usually it disperses as time goes on, namely
However, if this is not the case.
Proof. Let for a given . We argue indirectly and assume that, for every there exists such that
Since we get a contradiction. ∎
Thus if , by the above propositions the field and the condensed matter will not disperse, but will form bumps of matter which eventually might lead to the formation of one or more hylomorphic solitons.
3 The Nonlinear Klein-Gordon equation
The nonlinear Klein-Gordon equation (NKG) is given by
where , and satisfies
for some smooth function . Also we have used the notation
¿From now on, we always assume that
Under very mild assumptions on , the (NKG) admits spherical solitons. In this section, we recall some general facts about (NKG) and study the hylomorphic properties of solitons. In the next section we study the problem of existence and stability of solitons, and give a classification of the nonlinear terms in (NKG) according to the properties of the solutions it admits.
We recall the pioneering paper of Rosen  and , , . In Physics, the spherically symmetric solitary waves have been called -balls by Coleman in  and this is the name used in the Physics literature.
3.1 General features of NKG
In this case, the fields are functions with values in and equation (NKG) is the Euler-Lagrange equation of the action functional
where the Lagrangian density is given by
Moreover the nonlinear Klein-Gordon equation admits a formulation as an infinite dimensional Hamiltonian dynamical system. The phase space is given by and the state of the system at the time is then defined by a point , where is an admissible function for the functional (3.1) and . Equation (NKG) written as a first order system on takes the form
Hence, looking at as the variable conjugate to , the dynamics on is an infinite dimensional Hamiltonian system with the Hamiltonian function given by
The nonlinear Klein-Gordon equation is the simplest equation invariant for the Poincaré group and the action of on given by
By Noether theorem the existence of conservation laws implies the existence of integrals of motion. In particular, the time invariance of the Lagrangian implies the preservation of the energy given by
whereas the invariance with respect to the -action (3.3) implies the preservation of the hylomorphic charge given by
In the analysis of the behaviour of solutions of (NKG), it turns useful to write in polar form, namely
where and . Letting , and , a state is uniquely determined by the quadruple . Using these variables, the action (3.1) takes the form
and equation (NKG) becomes
In this form, energy and hylomorphic charge become
We now describe a possible interpretation of the hylomorphic quantities such as the hylomorphic charge, the hylomorphy ratio etc. The equation (NKG) describes how the density of hylomorphic charge moves with the dynamics. Indeed, looking at (NKG) in polar form, one notices that (3.6) is the continuity equation for the density of the hylomorphic charge (see (2.7) and (3.8)). When is negative, it can be interpreted as the density of “antiparticles”. ¿From this point of view, equation (3.6) describes the conservative transport of the hylomorphic charge by a particle field and the quantity defined in (2.2) can be considered as the rest energy of each particle.
In this interpretation, represents the total number of particles counted algebraically (particles minus antiparticles), represents the average energy of each particle, and the hylomorphy ratio defined in (2.4) is a dimensionless quantity which normalise the average energy for particle with respect to the rest energy . If , the average energy of each particle is larger than the rest energy; if , the opposite occurs and this fact means that particles interact with one another by an attractive force. In this interpretation, the volume occupied by the condensed matter defined in (2.9) consists of particles tied together. Moreover the quantity defined in (2.8), has the dimension of the energy and represents the binding energy of the particles. In fact, consider for example the case of identical particles. When the particles are free and at rest, their energy is given by the number of particles times the energy of a particle, namely , whereas the energy of the configuration is given by . Thus the energy necessary to “free” the particles is . Moreover represents the portion of the binding energy which is localised in the support .
Next we will compute the hylomorphy ratio for (NKG).
For (NKG) with of class the hylomorphy ratio defined in (2.4) takes the form
Proof. Let us write
where . We have to prove that the term defined in (2.2) satisfies .
Since by the classical Cauchy-Schwarz inequality
we have that
Then since for , it follows that
In order to prove the opposite inequality, consider the family of states , where
where we have again used for . ∎
This section is devoted to the existence of -balls and to the study of their structure.
4.1 Solitary waves for NKG
The easiest way to produce solitary waves of (NKG) consists in solving the static equation
and making a change of the frame of reference to give a velocity to the wave, setting for example
which is a solution of (NKG) which represents a bump travelling in the -direction with speed (we consider a normalisation of units of measure so that the speed of light is equal to 1).
Unfortunately Derrick, in the very well known paper , has proved that the request (which is necessary if we want the energy to be a non-negative invariant) implies that equation (4.1) has only the trivial solution. His proof is based on the Derrick-Pohozaev identity, which for (4.1) is given by
and the existence of non-trivial solutions of (4.4) is not prevented by the Derrick-Pohozaev identity, which in this case reads
Since the Lagrangian (3.2) is invariant for the Lorentz group, we can obtain other solutions just making a Lorentz transformation on it. Namely for , if we take the velocity , and set
it turns out that
is a solution of (NKG).
In particular, given a standing wave the function is a solitary wave which travels with velocity . Thus, if for example and is any solution of equation (4.4), then
is a solution of (NKG) provided that
Finally we remark that if is a solution of equation (4.4), then by a well known result on elliptic equations  it has necessarily spherical symmetry. Coleman called the spherically symmetric solitary waves of (NKG) Q-balls () and this is the name generally used in the Physics literature.
In order to prove the existence of solitons for (NKG), we are now led to study the existence of orbitally stable standing waves. Hence in particular we study existence results of couples which satisfy equation (4.4) under some general assumptions on the function .
4.2 Existence results for Q-balls
From now on we make the following assumptions on :
(W-i) (Positivity) ;
(W-ii) (Normalisation) ;
(W-iii) (Hylomorphy) .
Let us make some remarks on these assumptions.
(W-i) implies that the energy is positive for any state. Some aspects of the theory remain true weakening this assumption. In particular there are results in the case also for
since they are easier to prove with variational methods. However assumption (W-i) is required by most of the physical models and it simplifies some theorems.
(W-ii) is a normalisation condition. In order to have solitary waves it is necessary that There are results also in the null-mass case (see e.g.  and ). However the most interesting situations occur when In the latter case, re-scaling the independent variables , we may assume that .
(W-iii) is the crucial assumption. As we will see this assumption is necessary to have states with , hence hylomorphic solitons according to Definition 2.5.
Under the previous assumptions, we can write as
and the hylomorphy condition can be stated by saying that there exists such that . Actually, by our interpretation of (NKG) (see Section 3.1), is the nonlinear term which, when negative, produces an “attractive interaction” among the particles.
In  we prove that
Theorem 4.1 ().
If (W-i), (W-ii) and (W-iii) hold then (NKG) in , with , admits hylomorphic solitons of the form (4.4).
This result is recent in the form given here. In fact only recently it has been proved the orbital stability of the standing waves (4.3) with respect to the standard topology of and for all the which satisfy (W-ii), (W-iii) and a condition weaker than (W-i). Nevertheless Theorem 4.1 has a very long history starting with the pioneering paper of Rosen . Coleman  and Strauss  gave the first rigourous proofs of existence of solutions of the type (4.3) for some particular , and later necessary and sufficient existence conditions have been found by Berestycki and Lions .
where is the ground state solution of (4.4) for a fixed , and and are the energy and the hylomorphic charge of the standing wave . They prove that a necessary and sufficient condition for the orbital stability of the standing wave is the convexity of the map in . It is difficult to verify theoretically this condition for a general since cannot be computed explicitly. In some particular cases the properties of can be investigated, see for instance  where the authors give exact ranges of the frequency for which they obtain stability and instability of the respective standing waves for the nonlinear wave equation with as in (4.7). Moreover the computation of in a neighbourhood of a chosen can be extremely difficult due to the fact that the knowledge of is required.
The proof of Theorem 4.1 follows from some interesting preliminary results, which give the idea of the proof. We now sketch the main steps, following the general method exposed in Section 2.2, referring to  for details.
The first step is to define the following set
which corresponds to a subset of through the identification
Notice that the standing waves are contained in .
We introduce the functionals energy and charge on the space which with abuse of notation we still denote by and . They take the form
Without loss of generality we restrict ourselves to the case and . The relationship between the (NKG) equation and the energy on the space is given by the following proposition
Proposition 4.2 ().
The function is a standing wave for (NKG) if and only if is a critical point (with ) of the functional constrained to the manifold
Proof. A point is critical for constrained to the manifold if and only if there exists such that
Since , from the second equation , and the first becomes equation (4.4). ∎
This result gives a simple criterion to obtain standing waves for (NKG). Moreover we are interested in standing waves which are orbitally stable (see Definition 2.3), and to this aim we look for points of minimum of on . To this aim we use the notion of hylomorphy ratio, which on takes the form
One immediately gets
and if is defined as in (W-iii), then it is proved in  that
To prove the existence of point of minimum for the energy we use
Proposition 4.3 ().
If there exists such that , there exist points of minimum of the energy constrained to the manifold with .
A point of minimum corresponds to the standing wave , with a spherically symmetric function. Steps (S-2) and (S-3) follow from the fact that for isolated points of minimum the set of minimisers consists of the set
which has dimension . In the following we restrict ourselves to the case of isolated points of minimum, which is “generic” in the family of (NKG) equations.
is a Lyapunov function on . This implies that
Theorem 4.4 ().
If is a point of local minimum of the functional constrained to the manifold with , then is an orbitally stable standing wave.
Theorem 4.1 is a pure existence result and gives no information on the charge and frequency of the standing waves. In  we show that Proposition 4.3 implies that, under the assumptions (W-i), (W-ii) and (W-iii), there exists a threshold value such that for any there are hylomorphic solitons with hylomorphic charge . The existence of hylomorphic solitons for all hylomorphic charges can be obtained by a stronger version of assumption (W-iii). Let us consider the condition
(W-iv) (Behaviour at ) for small enough ( is defined in (4.8)).
Corollary 4.5 ().
If (W-i), (W-ii) and (W-iv) hold then for any there exists a hylomorphic soliton for (NKG) with hylomorphic charge .
We finish this section by giving a remark which is useful for the numerical approach to the existence of hylomorphic solitons for (NKG). We have found solitons as points of minimum for the energy on the manifold with fixed hylomorphic charge. Hence we are studying a minimisation problem in two variables with one constraint. It is immediate that this problem can be translated into a minimisation problem in one single variable with no constraints. We will use as independent variable the functions .
If is a minimiser of constrained to with fixed, then it is also a minimiser of constrained to . Moreover if , then
we can state Theorem 4.4 in the form
4.3 Numerical construction of Q-balls
In this section we provide basic details of our numerical method for constructing hylomorphic solitons. The numerical results are shown in the next section, where they illustrate a classification of (NKG) equations originally introduced therein.
Theorem 4.6 is straightforwardly exploited for the numerical construction of Q-balls: we fix a hylomorphic charge and look for points of minimum of the functional . Firstly, by the classical principle of symmetric criticality (see ), we can restrict ourselves to the analysis of radially symmetric functions , with in . We then consider the following evolutionary problem:
in which denotes a chosen upper bound for the domain (discussed below) and as in (4.16). The evolution of according to (4.18) is a gradient flow and therefore a non-increasing trend for is obtained, well suited as the sought minimisation process.
The problem (4.18) is then discretised by a classical line method; in particular, 2nd order and 1st order accurate finite differences have been respectively used for space and time discretisation (see e.g. ). The chosen charge is directly enforced at the -th time level (), by evaluating the frequency from the corresponding numerical solution through the discrete counterpart of (4.16). Moreover, time-advancing is stopped when , being a predefined threshold (a relative error on has been considered as well). The proposed method manages to efficiently converge by starting from several initial guesses: not only from Gaussian (Q-ball like) profiles but also from discontinuous ones (e.g. , with contained within the chosen domain and suitably set for obtaining the desired ). Finally, the domain extreme is chosen in such a way that it does not affect the numerical results (an a-posteriori check might be necessary: the chosen domain must be large enough to contain the soliton support, with some margin).
We remark that we did not try to implement a shooting method, on purpose. Indeed, sign-changing solutions are known to be unstable and, based on the given definition of soliton (implying stability), such solutions are of no interest in the present study. Conversely, the proposed numerical method guarantees to find a soliton, due to Theorem 4.6.
It is worth remarking that, once defined by means of the aforementioned strategy, it is easy to build moving Q-balls through the transformation (4.6); a two-dimensional (2D) example is shown in Figure 1.
4.4 Classification of the nonlinear terms
In this section we introduce four classes of behaviour for the nonlinear term , all classes admitting hylomorphic solitons. The classification is based on the existence and non-existence of solitons with small charge and with big norm for the (NKG) with in a fixed class. Moreover we state some general properties shared by hylomorphic solitons. Without any loss of generality we consider the case and .
Theorem 4.7 (Admissible frequencies).
Let (W-i), (W-ii) and (W-iii) hold and be a hylomorphic soliton for (NKG) in . Set
where is defined as in (W-iii). Then and there exists such that the frequency satisfies
Proof. We first show that and .
We recall that if is a hylomorphic soliton for (NKG), then its radial part satisfies equation (4.4). Hence using the Derrick-Pohozaev identity (4.5) for it follows that the energy (4.9) and the hylomorphy ratio (4.12) can be written as
By imposing that , a simple computation implies the first part of the theorem.
Again by imposing that we obtain that . ∎
Figure 2 shows a “typical” behaviour of the frequency and the hylomorphy ratio of a hylomorphic soliton as the hylomorphic charge varies. The frequency and the hylomorphy ratio are decreasing functions of the hylomorphic charge. We remark that as expected we find for all hylomorphic solitons.
Let satisfy (W-i), (W-ii) and (W-iii). We say that equation (NKG) (or W) is of type () if there exists such that any hylomorphic soliton of the form (4.3) satisfies
Assume that satisfies (W-i), (W-ii) and (W-iii), and write it as in (4.8). If
then equation (NKG) is of type ().
Proof. Let be a hylomorphic soliton for (NKG) and let us assume that there exists such that