On the meaning of the VakhitovKolokolov stability criterion for the nonlinear Dirac equation
Abstract
We consider the spectral stability of solitary wave solutions to the nonlinear Dirac equation in any dimension. This equation is wellknown to theoretical physicists as the Soler model (or, in one dimension, the GrossNeveu model), and attracted much attention for many years. We show that, generically, at the values of where the VakhitovKolokolov stability criterion breaks down, a pair of real eigenvalues (one positive, one negative) appears from the origin, leading to the linear instability of corresponding solitary waves.
As an auxiliary result, we state the Virial identities (“Pohozhaev theorem”) for the nonlinear Dirac equation. We also show that are the eigenvalues of the nonlinear Dirac equation linearized at , which are embedded into the essential spectrum as long as . This result holds for the nonlinear Dirac equation with any nonlinearity of the Soler form (“scalarscalar interaction”) and in any dimension.
As an illustration of the spectral stability methods, we revisit Derrick’s theorem and sketch the VakhitovKolokolov stability criterion for the nonlinear Schrödinger equation.
1 Introduction
Field equations with nonlinearities of local type are natural candidates for developing tools which are then used for the analysis of systems of interacting equations. Equations with local nonlinearities have been appearing in the Quantum Field Theory starting perhaps since fifties [Sch51a, Sch51b], in the context of the classical nonlinear meson theory of nuclear forces. The nonlinear version of the Dirac equation is known as the Soler model [Sol70]. The existence of standing waves in this model was proved in [Sol70, CV86]. Existence of localized solutions to the DiracMaxwell system was addressed in [Wak66, Lis95] and finally was proved in [EGS96] (for ) and [Abe98] (for ). The local wellposedness of the DiracMaxwell system was considered in [Bou96]. The local and global wellposedness of the Dirac equation was further addressed in [EV97] (semilinear Dirac equation in ), [Bou00] (Dirac – KleinGordon system in ), and in [MNNO05] (nonlinear Dirac equation in ). The question of stability of solitary wave solutions to the nonlinear Dirac equation attracted much attention for many years, but only partial numerical results were obtained; see e.g. [AC81, AKV83, AS83, AS86, Chu07]. The analysis of stability with respect to dilations is performed in [SV86, CKMS10].
Understanding the linear stability is the first step in the study of stability properties of solitary waves. Absence of an eigenvalue with a positive real part will be referred to as the spectral stability, while its absence as the spectral (or linear) instability. After the spectrum of the linearized problem for the nonlinear Schrödinger equation [VK73] was understood, the linearly unstable solitary waves can be proved to be ( “nonlinearly”, or “dynamically”) unstable [Gri88, GO10], while the linearly stable solitary waves of the nonlinear Schrödinger and KleinGordon equations [Sha83, SS85, Wei86] and more general invariant systems [GSS87] were proved to be orbitally stable. The tools used to prove orbital stability break down for the Dirac equation since the corresponding energy functional is signindefinite. On the other hand, one can hope to use the dispersive estimates for the linearized equation to prove the asymptotic stability of the standing waves, similarly to how it is being done for the nonlinear Schrödinger equation [Wei85], [SW92], [BP93], [SW99], and [Cuc01]. The first results on asymptotic stability for the nonlinear Dirac equation are already appearing [PS10, BC11], with the assumptions on the spectrum of the linearized equation playing a crucial role.
In this paper, we study the spectrum of the nonlinear Dirac equation linearized at a solitary wave, concentrating on bifurcation of real eigenvalues from .
Derrick’s theorem
As a warmup, let us consider the linear instability of stationary solutions to a nonlinear wave equation,
(1.1) 
We assume that the nonlinearity is smooth. Equation (1.1) is a Hamiltonian system, with the Hamiltonian
There is a wellknown result [Der64] about nonexistence of stable localized stationary solutions in dimension (known as Derrick’s Theorem). If is a localized stationary solution to the Hamiltonian equations , , then, considering the family , one has and then it follows that as long as . That is, for a variation corresponding to the uniform stretching, and the solution is to be unstable. Let us modify Derrick’s argument to show the linear instability of stationary solutions in any dimension.
Lemma 1.1 (Derrick’s theorem for ).
For any , a smooth finite energy stationary solution to the nonlinear wave equation is linearly unstable.
Proof.
Since satisfies , we also have . Due to , vanishes somewhere. According to the minimum principle, there is a nowhere vanishing smooth function (due to being elliptic) which corresponds to some smaller (hence negative) eigenvalue of , , with . Taking , we obtain the linearization at , , which we rewrite as
Let us also mention that , showing that is not positivedefinite. ∎
Remark 1.2.
A more general result on the linear stability and (nonlinear) instability of stationary solutions to (1.1) is in [KS07]. In particular, it is shown there that the linearization at a stationary solution may be spectrally stable when this particular stationary solution is not from (such examples exist in higher dimensions).
VakhitovKolokolov stability criterion for the nonlinear Schrödinger equation
To get a hold of stable localized solutions, Derrick suggested that elementary particles might correspond to stable, localized solutions which are periodic in time, rather than timeindependent. Let us consider how this works for the (generalized) nonlinear Schrödinger equation in one dimension,
(1.2) 
where is a smooth function with . One can easily construct solitary wave solutions , for some and : satisfies the stationary equation , and can be chosen strictly positive, even, and monotonically decaying away from . The value of can not exceed . We consider the Ansatz , with . The linearized equation on is called the linearization at a solitary wave:
(1.3) 
with
(1.4) 
Note that since , the action of on considered as taking values in is linear but not linear. Since , the essential spectrum of and is .
First, let us note that the spectrum of is located on the real and imaginary axes only: . To prove this, we consider is positivedefinite (, being nowhere zero, corresponds to its smallest eigenvalue), we can define the selfadjoint root of ; then Since
with the inclusion due to being selfadjoint. Thus, any eigenvalue satisfies .
Given the family of solitary waves, , , we would like to know at which the eigenvalues of the linearized equation with appear. Since , such eigenvalues can only be located on the real axis, having bifurcated from . One can check that belongs to the discrete spectrum of , with
for all which correspond to solitary waves. Thus, if we will restrict our attention to functions which are even in , the dimension of the generalized null space of is at least two. Hence, the bifurcation follows the jump in the dimension of the generalized null space of . Such a jump happens at a particular value of if one can solve the equation . This leads to the condition that is orthogonal to the null space of the adjoint to , which contains the vector ; this results in . A slightly more careful analysis [CP03] based on construction of the moving frame in the generalized eigenspace of shows that there are two real eigenvalues that have emerged from when is such that becomes positive, leading to a linear instability of the corresponding solitary wave. The opposite condition,
(1.5) 
is the VakhitovKolokolov stability criterion which guarantees the absence of nonzero real eigenvalues for the nonlinear Schrödinger equation. It appeared in [VK73, Sha83, GSS87] in relation to linear and orbital stability of solitary waves. The above approach fails for the nonlinear Dirac equation since is no longer positivedefinite.
For the completeness, let us present a more precise form of the VakhitovKolokolov stability criterion [VK73].
Lemma 1.3 (VakhitovKolokolov stability criterion).
There is , , where is the linearization (1.3) at the solitary wave , if and only if at this value of .
Proof.
We follow [VK73]. Assume that there is , . The relation implies that . It follows that is orthogonal to the kernel of the selfadjoint operator (which is spanned by ):
hence there is such that and . Thus, the inverse to can be applied: . Then
Since is positivedefinite and , it follows that . Since , , therefore the quadratic form is not positivedefinite on vectors orthogonal to . According to Lagrange’s principle, the function corresponding to the minimum of under conditions and satisfies
(1.6) 
Since , we need to know whether could be negative. Since , one has . Due to vanishing at one point (), there is exactly one negative eigenvalue of , which we denote by . (This eigenvalue corresponds to some nonvanishing eigenfunction.) Note that , or else would have to be equal to , with the corresponding eigenfunction of , but then , having to be nonzero, could not be orthogonal to . Denote . Let us consider , which is defined and is smooth for . (Note that is defined for since the corresponding eigenfunction is odd while is even.) If , then, by (1.6), we would have , and since , one has . On the other hand,
Alternatively, let . We consider the function , . Since , , and (where is the smallest eigenvalue of ), there is such that . Then we define . Since , there is such that . It follows that the quadratic form is not positive definite:
Thus, there is such that ; then also . Let be the corresponding eigenvector, ; then hence . ∎
Our conclusions:

Point eigenvalues of the linearized Dirac equation may bifurcate (as changes) from the origin, when the dimension of the generalized null space jumps up (when the VakhitovKolokolov criterion breaks down).

Since the spectrum of the linearization does not have to be a subset of , there may also be point eigenvalues which bifurcate from the imaginary axis into the complex plane. (We do not know particular examples of such behavior for the nonlinear Dirac equation.)

Moreover, there may be point eigenvalues already present in the spectra of linearizations at arbitrarily small solitary waves. Formally, we could say that these eigenvalues bifurcate from the essential spectrum of the free Dirac operator (divided by ), which can be considered as the linearization of the nonlinear Dirac equation at the zero solitary wave.
In the present paper we investigate the first scenario. The main result (Lemma 4.1) states that if the VakhitovKolokolov breaks down at some point , then, generically, the solitary waves with from an open onesided neighborhood of are linearly unstable.
2 Linearization of the nonlinear Dirac equation
The nonlinear Dirac equation in has the form
(2.1) 
where , is even and smooth, with . The Dirac matrices and satisfy the relations
where is an unit matrix. We will always assume that , we assume ; in the case , one could take are the standard Pauli matrices. Equation (2.1), usually with , is called the Soler model [Sol70], which has been receiving a lot of attention in theoretical physics in relation to classical models of elementary particles. , where . In the case
Remark 2.1.
In terms of the Dirac matrices, equation (2.1) takes the explicitly relativisticallyinvariant form where , , , .
Definition 2.2.
The solitary waves are solutions to (2.1) of the form , , .
Below, we assume that there are solitary waves for from some nonempty set :
(2.2) 
with smoothly depending on .
We will not indicate the dependence on explicitly, and will write instead of .
The profile of a stationary wave satisfies the stationary nonlinear Dirac equation
(2.3) 
The energy and charge functionals corresponding to the nonlinear Dirac equation (2.1) are given by
where is the antiderivative of which satisfies . is the charge functional which is (formally) conserved for solutions to (2.1) due to the invariance. The nonlinear Dirac equation (2.1) can be written in the Hamiltonian form as
(2.4) 
where the primes denote the Fréchét derivative of the functionals , with respect to .
Let us write the solution in the form . The linearized equation on is given by
(2.5) 
where corresponds to a multiplication by and
Note that, because of the presence of , the action of on is linear but not linear. Because of this, it is convenient to write it as an operator acting on vectors from ; then (2.5) takes the following form:
(2.6) 
where , and
(2.7) 
Note that , , and correspond to multiplication by , , and under the correspondence.
Remark 2.3.
When , one can take (so that ). Then has a particularly simple form since can be chosen valued in : , . The numerical and analytical study of spectra of , in this case is contained in [BC09]. , with
Lemma 2.4.
.
Proof.
One has . Note also that has a spectrum . Taking into account that the symbol of at is one concludes that Since the eigenvalues of are , corresponding to clock and counterclockwise rotations in , one deduces that . ∎
Lemma 2.5.
The null space of is given by
Proof.
Lemma 2.6.
Let be an hermitian matrix anticommuting with , , and with . Then is an eigenfunction of and of , corresponding to the eigenvalue .
Remark 2.7.
If , one can take .
Proof.
Since anticommutes with () and with , and taking into account (2.3), we have:
Since and are Hermitian, therefore, one also has ∎
It follows that the linearization operator has an eigenvalue :
Since is symmetric with respect to and , for any in (2.1) and in any dimension , we have:
Corollary 2.8.
are eigenvalues of .
Remark 2.9.
For , the eigenvalues are embedded in the essential spectrum. This is in contradiction with the Hypothesis (H:6) in [BC11] on the absence of eigenvalues embedded in the essential spectrum, although we hope that this difficulty could be dealt with using a minor change in the proof.
Remark 2.10.
The result of Corollary 2.8 takes place for any nonlinearity and in any dimension. The spatial dimension and the number of components of could be such that there is no matrix which anticommutes with , , and with ; then the eigenvector corresponding to can be constructed using the spatial reflections.
3 Virial identities
When studying the bifurcation of eigenvalues from , we will need some conclusions about the generalized null space of the linearized operator. We will draw these conclusions from the Virial identities, which are also known as the Pohozhaev theorem [Poh65]. In the context of the nonlinear Dirac equations, similar results were presented in [ES95].
Lemma 3.1.
For a differentiable family , , one has
Proof.
This immediately follows from (2.4). ∎
We split the Hamiltonian into where
Lemma 3.2 (Pohozhaev Theorem for the nonlinear Dirac equation).
For each solitary wave , there are the following relations:
Proof.
Remark 3.3.
For all nonlinearities for which the existence of solitary wave solutions is proved in [ES95], one has for all except finitely many points (e.g. ); hence for these solitary waves one has . In particular, for , one has .
4 Bifurcations from
Lemma 4.1.
Assume that the nonlinearity satisfies the following inequality (see Remark 3.3):
(4.1) 
Further, assume that and are spherically symmetric and that
If , then the generalized null space of is given by
If vanishes at , then . Moreover, generically, there is an eigenvalue with for from an open onesided neighborhood of .
Remark 4.2.
Proof.
Taking the derivative of (2.3) with respect to , we get
(4.2) 
Since , we have
(4.3) 
Similarly, since ,
(4.4) 
Using (4.3) and (4.4), we have
(4.5) 
Until the end of this section, it will be more convenient for us to work in terms of (see (2.7)). We summarize the above relations (2.8), (2.9), (4.2), (4.3), (4.4), and (4.5) as follows:
where
with , . There are no such that Indeed, checking the orthogonality of with respect to , we have: