SPECTRAL STABILITY OF BI-FREQUENCY SOLITARY WAVES IN SOLER AND DIRAC–KLEIN–GORDON MODELS

A BSTRACT . We construct bi-frequency solitary waves of the nonlinear Dirac equation with the scalar self-interaction, known as the Soler model (with an arbitrary nonlinearity and in arbitrary dimension) and the Dirac–Klein–Gordon with Yukawa self-interaction. These solitary waves provide a natural implementation of qubit and qudit states in the theory of quantum computing. We show the relation of ± 2 ω i eigenvalues of the linearization at a solitary wave, Bogoliubov SU (1 , 1) symmetry, and the existence of bi-frequency solitary waves. We show that the spectral stability of these waves reduces to spectral stability of usual (one-frequency) solitary waves.

which is also based on the quantityψψ): where M > 0 is the mass of the scalar field Φ(t, x) ∈ R.
The solitary wave solutions in the Soler model (already constructed in [29]) possess certain stability properties [3,13,6]; in particular, small amplitude solitary waves corresponding to the nonrelativistic limit ω m of the charge-subcritical and charge-critical case f (τ ) = |τ | k with k 2/n are spectrally stable: the linearized equation on the small perturbation of a particular solitary wave has no exponentially growing modes. The opposite situation, the linear instability of small amplitude solitary waves (presence of exponentially growing modes) in the charge-supercritical case k > 2/n was considered in [11]. Recent results on asymptotic stability of solitary waves in the nonlinear Dirac equation [7,26,10] rely on the assumptions on the spectrum of the linearization at solitary waves, although this information is not readily available, especially in dimensions above one. This stimulates the study of the spectra of linearizations at solitary waves. It was shown in [3] that the Soler model in one spatial dimension linearized at a solitary wave φ(x)e −iωt has eigenvalues ±2ωi. While the zero eigenvalues correspond to symmetries of the system (unitary, translational, etc.), the eigenvalues ±2ωi are related to the presence of bi-frequency solitary waves and to the Bogoliubov SU(1, 1) symmetry of the Soler model and Dirac-Klein-Gordon models, first noticed by Galindo in [16]. In the three-dimensional case (n = 3, N = 4) with the standard choice of the Dirac matrices, this symmetry group takes the form where K : C N → C N is the antilinear operator of complex conjugation; the group isomorphism is given by a − ibγ 2 K → a b bā . By the Noether theorem, the continuous symmetry group leads to conservation laws (see Section 2). The Bogoliubov group, when applied to standard solitary waves φ(x)e −iωt in the form of the Wakano Ansatz [31], φ(x) = v(r, ω)ξ iu(r, ω) x · σ |x| ξ , ξ ∈ C 2 , |ξ| = 1, produces bi-frequency solitary waves of the form aφ(x)e −iωt + bφ C (x)e iωt , a, b ∈ C, |a| 2 − |b| 2 = 1, with φ C = −iγ 2 Kφ the charge conjugate of φ; here −iγ 2 K is one of the infinitesimal generators of SU(1, 1). Above, v(r, ω) and u(r, ω) are real-valued functions which satisfy with in the case of nonlinear Dirac equation (1) (see e.g. [5]) and in the case of Dirac-Klein-Gordon system (2). We assume that the functions u(r, ω) and v(r, ω) satisfy sup r≥0 |u(r, ω)/v(r, ω)| < 1, which is true in particular for small amplitude solitary waves with ω m; see e.g. [5].
We now switch to the general case of a general spatial dimension n ≥ 1 and a general number of spinor components N ≥ 2. By usual arguments (see e.g. [5]), without loss of generality, we may assume that the Dirac matrices have the form Here σ j , 1 ≤ j ≤ n, are N 2 × N 2 matrices which are the higher-dimensional analogue Pauli matrices: In the case n = 3, N = 4, one takes σ j , 1 ≤ j ≤ 3, to be the standard Pauli matrices.
Remark 1.1. In general, σ j are not necessarily self-adjoint; for example, for n = 4 and N = 4, one can choose σ j to be the standard Pauli matrices for 1 ≤ j ≤ 3 and set σ 4 = iI 2 .
We denote In Section 3, we show that if φ(x)e −iωt , with φ from (4), is a solitary wave solution to (1) (or (2)), then there is the following family of exact solutions to (1) (or (2), respectively): This shows that in any dimension there is a larger symmetry group, SU(N/2, N/2), which is present at the level of bi-frequency solitary wave solutions in the models (1) and (2) while being absent at the level of the Lagrangian.
Two-frequency solitary waves (13) clarify the nature of the eigenvalues ±2ωi of the linearization at (one-frequency) solitary waves in the Soler model: these eigenvalues could be interpreted as corresponding to the tangent vectors to the manifold of bi-frequency solitary waves. See Corollary 3.2 below. We point out that the exact knowledge of the presence of ±2ωi eigenvalues in the spectrum of the linearization at a solitary wave is important for the proof of the spectral stability: namely, it allows us to conclude that in the nonrelativistic limit ω m the only eigenvalues that bifurcate from the embedded thresholds at ±2mi are ±2ωi; no other eigenvalues can bifurcate from ±2mi, and in particular no eigenvalues with nonzero real part. For details, see [6].
We point out that the asymptotic stability of standard, one-frequency solitary waves can only hold with respect to the whole manifold of solitary wave solutions (13), which includes both one-frequency and bi-frequency solitary waves: if a small perturbation of a onefrequency solitary wave is a bi-frequency solitary wave, which is an exact solution, then convergence to the set of one-frequency solitary waves is out of question. In this regard, we recall that the asymptotic stability results [26,10] were obtained under certain restrictions on the class of perturbations. It turns out that these restrictions were sufficient to remove not only translations, but also the perturbations in the directions of bi-frequency solitary waves; this is exactly why the proof of asymptotic stability of the set of one-frequency solitary waves with respect to such class of perturbations was possible at all.
While the stability of one-frequency solitary waves turns out to be related to the existence of bi-frequency solitary waves, one could question the stability of such bi-frequency solutions, too. In Section 4, we show that the bi-frequency solitary waves are spectrally stable as long as so are the corresponding one-frequency solitary waves. While this conclusion may seem natural, we can only give the proof for the case when the number of spinor components satisfies N ≤ 4 (which restricts the spatial dimension to n ≤ 4).
Let us mention that the bi-frequency solitary waves (5) may play a role in Quantum Computing. Indeed, such states produce a natural implementation of qubit states a|0 +b|1 , |a| 2 + |b| 2 = 1, except that now the last relation takes the form |a| 2 − |b| 2 = 1. Just like for standard cubits, our bi-frequency states (5) have two extra parameters besides the orbit of the U(1)-symmetry group. Below, we are going to show that qubits (5) can be linearly stable. Moreover, the manifold of bi-frequency solitary waves (13) admits a symmetry group SU(N/2, N/2) (which may be absent on the level of the Lagrangian). For these solitary waves, the number of degrees of freedom (after we factor out the action of the unitary group) is d = 2N − 2. The states (13) with N ≥ 4 correspond to higher dimensional versions of qubits -d-level qudits, quantum objects for which the number of possible states is greater than two. These systems could implement quantum computation via compact higher-level quantum structures, leading to novel algorithms in the theory of quantum computing. Bi-frequency solitary waves could also provide a simple stable realization of higher-dimensional quantum entanglement, or hyperentanglement, which is used in cryptography based on quantum key distribution; by [8,14], using qudits over qubits provides increased coding density for higher security margin and also an increased level of tolerance to noise at a given level of security. Qudits have already been implemented in the system with two electrons [23] as quantum walks of several electrons (just like one qubit could be represented by a quantum walk, a distribution of an electron in a "quantum tunnel" between individual quantum dots, considered as potential wells). Being sensitive to the external noise, the quantum-walk implementation of qudits in [23] was indicated to be highly unstable, requiring excessive cooling and making the practical usage very difficult. In [19], the on-chip implementation of qudit states is achieved by creating photons in a coherent superposition of multiple high-purity frequency modes. We point out that the bi-frequency solitary waves (13) can possess stability properties, as we show below; moreover, the simplicity of the model suggests that such states could be implemented using photonic states in optical fibers without excessive quantum circuit complexity.
We need to mention that several novel nonlinear photonic systems currently explored are modeled by Dirac-like equations (often called coupled mode systems) which are similar to (1). Examples include fiber Bragg gratings [15], dual-core photonic crystal fibers [4], and discrete binary arrays, which refer to systems built as arrays coupled of elements of two types. Earlier experimental work on binary arrays has already shown the formation of discrete gap solitons [24]. Three of the many novel examples that have been recently considered are: a dielectric metallic waveguide array [2,1]; an array of vertically displaced binary waveguide arrays with longitudinally modulated effective refractive index [22], and arrays of coupled parity-time (PT ) nanoresonators [20]. We also constructed bi-frequency solutions of the form (5) in the Dirac-type models with the PT -symmetry which arise in nonlinear optics in the model describing arrays of optical fibers with gain-loss behavior [12]. This venue of research is pursued for optical implementation of traditional circuits [28,27] aimed at the energy-efficient computing and at the challenges in reducing the footprint of optics-based devices.
2. The Bogoliubov SU(1, 1) symmetry and associated charges. Chadam and Glassey [9] noted an interesting feature of the model (2): under the standard choice of 4 × 4 Dirac matrices α j and β, as long as the solution ψ is sufficiently regular, there is a conservation of the quantity As a consequence, if (14) is zero at some and hence at all moments of time, then |ψ 1 | = |ψ 4 | and |ψ 2 | = |ψ 3 | for almost all x and t, hence ψ * βψ ≡ 0 (in the distributional sense), meaning that the self-interaction plays no role in the evolution and that as the matter of fact the solution solves the linear equation (without self-interaction). A similar feature of the Soler model (1) was analyzed in [25]. The relation of the conservation of the quantity (14) in Dirac-Klein-Gordon to the SU(1, 1) symmetry of the corresponding Lagrangian based on combinations ofψγ 0 D m ψ andψψ was noticed by Galindo [16]. Let us state the above results in a slightly more general setting. Assume that B ∈ End (C N ) is a matrix which satisfies The relations (15) imply that Above, "t" denotes the transpose.
Proof. The first statement is an immediate consequence of the fact that BK commutes with the flow of the equation. If ψ(t, x) satisfies iψ = D m ψ − f (ψ * βψ)βψ, then Finally, since BKψ = zψ, then We took into account that (βB) t = −βB by (16).
As in [16], the Lagrangians of the Soler model (1) and of the Dirac-Klein-Gordon model (2), with the densities are invariant under the action of the continuous symmetry group g ∈ G Bogoliubov , g : ψ → (a + bBK)ψ, |a| 2 − |b| 2 = 1 (cf. (3)). The Noether theorem leads to the conservation of the standard charge Q = R 3 ψ * ψ dx corresponding to the standard charge-current densityψγ µ ψ (note that the unitary group is a subgroup of SU(1, 1)), and the complex-valued Bogoliubov charge Λ = R n ψ * BKψ dx which corresponds to the complex-valued four-current density ψ * γ 0 γ µ BKψ. Now Galindo's observation [16] could be stated as follows. Lemma 2.3 (The Bogoliubov SU(1, 1) symmetry and the charge conservation).

For solutions of the nonlinear Dirac equation
Taking the linear combination with (1), we arrive at It remains to notice that ϕ * Kρ = (Kϕ) * ρ, ∀ϕ, ρ ∈ C N , hence The invariance of the Hamiltonian density follows from (cf. (18)) By the Nöther theorem, the invariance under the action of a continuous group results in the conservation laws. Let us check the (formal) conservation of the complex-valued Λ-charge. Writing f = f (ψ * βψ), we have: In the last relation, we took into account the anticommutation relations from (15). We also note that for the densities, we have showing that the Minkowski vector of the Bogoliubov charge-current density is given by Remark 2.4. Three conserved quantities, one being real and one complex, correspond to dim R SU(1, 1) = 3.
Remark 3.3. The presence of the eigenvalues ±2ωi in the spectrum of the linearization of the Soler model at a solitary wave was noticed in [3] (initially in the one-dimensional case) and eventually led to the conclusion that there exist bi-frequency solitary waves. We note that the existence of such bi-frequency solutions could have already been deduced applying the Bogoliubov transformation (3) from [16] to one-frequency solitary waves φ(x)e −iωt which were constructed in [29].
We define the solitary manifold of one-and bi-frequency solutions of the form (24) corresponding to some value ω by In general, the solitary manifold M ω can be larger than the orbit of φe −iωt under the action of the available symmetry groups: G Bogoliubov defined in (3) and SO(n); we denote this orbit by In lower spatial dimensions n ≤ 2, when N = 2, the orbit O φ is given by with G Bogoliubov = a + bσ 1 K, a, b ∈ C, |a| 2 − |b| 2 = 1 ; thus, O φ coincides with the solitary manifold In three spatial dimensions, n = 3 and N = 4, the solitary manifold M ω is larger than the orbit O φ of φ(x)e −iωt under the action of the available symmetry groups: spatial rotations SO(n) and the Bogoliubov group G Bogoliubov given by elements of the form Note that in the above inequality for dim R O φ one has "strictly smaller", since the generator corresponding to the standard U(1)-invariance which enters the Lie algebra of G Bogoliubov also coincides with the generator of rotation around z-axis.
In the case n = 4, N = 4, the symmetry group simplifies to U(1) since the generator −iγ 2 K is no longer available: the Dirac operator now contains −iα 4 ∂ x 4 = βγ 5 ∂ x 4 (with , which breaks the anticommutation of D m with −iγ 2 K: with the Lie algebras of SO(4) and G Bogoliubov sharing one element (a generator of the standard U(1)-symmetry). Moreover, the action of SO(4) (dim R = 6) on C 2 (dim R = 4) could not be faithful; the orbit of an element ξ ∈ C 2 under the action of SO(4) is only three-dimensional. As a result, in the case n = 4, N = 4, one has Remark 3.5. We can rephrase the above situation in the following way. When moving from n = 3 to n = 4, additional rotations in R 4 do not add to the orbit of ξ ∈ C 2 which has already been of maximal dimension when n = 3 (which equals three: it is the real dimension of the unit sphere in C 2 ), while the loss of the generator BK from the Bogoliubov group led to the loss of two real dimensions of the orbit O φ .
Remark 3.6. Let us briefly discuss the pseudo-scalar theories in spatial dimension n = 3.
where we take (t, x) = e iωt g −1 J j=1 ρ j (x)e Λj t , with g −1 =ā − bBK. The above relation shows that the exponential growth of J j=1 ρ j (x)e Λj t is in one-to-one correspondence to the exponential growth of (t, x). As a result, the spectral stability of the bifrequency wave aφ(x)e −iωt + bχ(x)e iωt (cf. Definition 4.1) takes place if and only if the corresponding one-frequency solitary wave φ(x)e −iωt is spectrally stable. This completes the proof in the case n ≤ 2, N = 2.
Now we assume that n ≤ 4, with a = |Ξ|, b = |H|, ξ = Ξ/|Ξ| and η = H/|H| (with φ ξ and χ η from (23) and (25), respectively) be a bi-frequency solitary wave. We will consider the perturbation of this solitary wave in the form where we will impose the following condition on ρ(t, x) and σ(t, x): If (29) is satisfied, we will say that there is no frequency mixing; in this case, ψ * βψ does not contribute terms with factors e −2iωt or e 2iωt . We will show that indeed there is a way to split the perturbation into ρ(t, x) and σ(t, x) so that (29) is satisfied (see Proposition 4.3 and Remark 4.6 below).
Let (e j ) 1≤j≤N/2 be the standard basis in C N/2 and let R, S ∈ SU(N/2) be such that ξ = Re 1 , η = Se 1 . Denote (Above, we do not indicate the dependence of v, u of ω.) We consider the perturbation of the bi-frequency solitary wave (28) in the form Above, p j , q j , r j , s j , 1 ≤ j ≤ N/2, are complex scalar-valued functions of x and t. The condition (29) of the absence of frequency mixing takes the form As long as (32) is satisfied, the linearized terms in the expansion of ψ * βψ do not contain factors e ±2iωt ; the ones that are left are given by The linearized equation will contain two groups of terms, with factors e ±iωt ; to satisfy the linearized equation, it is enough to equate these groups separately. The terms with the factor e −iωt : the terms with the factor e iωt : Remark 4.2. When deriving the above equations, we eliminated terms with the derivatives of φ j and χ j by using the stationary Dirac equations satisfied by φ j and χ j : Proof. We claim that if p j , q j , r j , and s j , 1 ≤ j ≤ 2, satisfy (36), then equations (34) and (35) yield Multiplying the relation (34) by β and coupling with φ j (in the C N -sense; no integration in x): We took into account that φ * j βχ k = 0 for all 1 ≤ j, k ≤ N/2 (cf. Lemma 4.4 below). Multiplying the relation (35) by β and coupling with χ j : Multiplying the relation (34) by β and coupling with χ j one has: Multiplying the relation (35) by β and coupling with φ j : The proof of Proposition 4.3 will follow if we prove that (38) and (39) are complex conjugates of each other, and that so are (40) and (41).
To argue that the last two lines are anti-complex conjugates, we note that and then use the same reasoning as above (when showing that (45) and (46) are complex conjugates, basing on the identities (47)). Remark 4.6. We note that the relation (32) is satisfied in the invariant subspace described in Proposition 4.3, and thus there is no frequency mixing in this subspace: given ψ of the form (31) with r j =q j , s j =p j , 1 ≤ j ≤ N/2, the expression ψ * βψ does not contain terms with the factors e ±2iωt .
We introduce the following functions: at each x ∈ R n , these functions form a basis in C N .
Proof. For any 1 ≤ j ≤ N/2 and any in the last equality, one can use the Schur complement to compute the determinant of a matrix written in the block form. Using (9) and taking into account that σ * r σ r = I 2 , one concludes that the right-hand side of the above is separated from zero uniformly in x ∈ R n .
The perturbation of a bi-frequency solitary wave could be rewritten as follows (cf. (31)): Taking into account (49), we note that P j and Q j in the above formula differ from p j and q j in (31) by the factor of v(r): We claim that at the initial moment there is a unique way to decompose the perturbation f ∈ L 2 (R n , C N ) over the terms in (50) with factors e ±iωt : Lemma 4.8. Let a, b ∈ C, |a| 2 − |b| 2 = 1, and let (9) be satisfied. Then for any ∈ L 2 (R n , C N ), there is a unique choice of scalar functions (P j , Q j ) 1≤j≤N/2 ∈ L 2 (R n , C) N such that (51) The map L 2 (R n , C N ) → L 2 (R n , C) N , → (P j , Q j ) 1≤j≤N/2 , is continuous.
Proof. Let ∈ L 2 (R n , C N ). By Lemma 4.7, there are f j ∈ L 2 (R n , C) and g j ∈ L 2 (R n , C) be such that = N/2 j=1 Φ j f j + N/2 j=1 X j g j , and the map L 2 (R n , C N ) → L 2 (R n , C) N , → (f j , g j ) 1≤j≤N/2 ∈ L 2 (R n , C) N is continuous. Equation (51) takes the form aP j (x) + bQ j (x) = f j (x), aQ j (x) + bP j (x) = g j (x), 1 ≤ j ≤ N/2. (52) Since |b/a| < 1, for any (f j , g j ) 1≤j≤N/2 ∈ L 2 (R n , C) N the map is a contraction in L 2 (R n , C) N and thus has a unique fixed point (a solution to (52)) which continuously depends on (f j , g j ).
We can now conclude the proof of We conclude that the bi-frequency solitary wave solution (28) to the nonlinear Dirac equation (1) (or the Dirac-Klein-Gordon system (2)) is linearly unstable if and only if the one-frequency solitary wave φ ξ (x)e −iωt is linearly unstable.