Soler model: stability, bi-frequency solitons, and SU(1,1) Andrew Comech, Texas A&M University and I.I.T.P. (Moscow) Nabile Boussa¨ıd, Universite´ Bourgogne – Franche-Comte,´ Besanc¸on, France July 2016
Einstein: E2 = p2 + m2 Schrodinger:¨ (i∂ )2ψ = ( i )2ψ + m2ψ t − ∇ 2 E = m2 + p2 m + p ≈ 2m p [Schrodinger¨ 26]: i∂ ψ = 1 ( i )2ψ t 2m − ∇ [Dirac28]: E = p2 + m2 = α p + βm, · p i∂ ψ = ( iα + βm) ψ, ψ(x,t) C4, x R3 t − ·∇ ∈ ∈ Dm | {z } 2 2 Dirac matrices α (1≤j≤3) and β: self-adjoint, ( iα + βm) = ∆+ m j − ·∇ − 0 σj I2 0 Standard choice: αj = (Pauli matrices), β = σj 0 0 I2 −
1 Dirac–Maxwell system ψ¯ := ψ∗β γµ(i∂ eA )ψ = mψ, ✷Aµ = eψγ¯ µψ, ∂ Aµ = 0. µ − µ µ
[Esteban et al.96, Abenda98]: solitary waves ψ(x,t) = φ(x)e−iωt, ω ( m,m) ∈ − Dirac–Maxwell and Dirac–Coulomb systems
40
30 −iωt Q(φωe ) 20
10
0 E(φ e−iωt) ω -10
-20 -1 -0.5 0 0.5 1 m<ω m − ≤ [Comech15]: solitary waves with ω = m for Dirac–Coulomb; φ(x) e−const√|x|. ∼ γ0 = β, γj = βαj , 1 j 3 ≤ ≤
2 Self-interacting spinors ψ¯ := ψ∗β Models of self-interacting spinor field: [Ivanenko38, Finkelstein et al.51, Finkelstein et al.56, Heisenberg57] [...]
Soler model [Ivanenko38, Soler70], scalar self-interaction: µ k+1 LSoler = ψ¯(iγ ∂ m)ψ + ψψ¯ k > 0 µ − | | in (1+1)D: massive Gross-Neveu model [Gross & Neveu74]
Massive Thirring model [Thirring58], vector self-interaction:
k+1 µ µ 2 LMTM = ψ¯(iγ ∂ m)ψ + ψγ¯ ψ ψγ¯ ψ k > 0 µ − µ
J µ(x) = ψ¯(x)γµψ(x): “charge-current density”
3 Soler model: NLD with scalar self-interaction
i∂ ψ = ( iα + mβ) ψ (ψψ¯ )kβψ, ψ(x,t) CN , x Rn t − ·∇ − ∈ ∈ Dm | {z } [Soler70, Cazenave & Vazquez´ 86]: existence of solitary waves in R3, •
ψ(x,t) = φ (x)e−iωt, ω (0,m), φ H1(R3) ω ∈ ω ∈
Attempts at stability: [Bogolubsky79, Alvarez & Soler86, Strauss & Vazquez´ 86] ... • Numerics suggest that (all?) solitary waves in 1D cubic Soler model are stable: • [Alvarez & Carreras81, Alvarez & Soler83, Cooper et al.10, Berkolaiko & Comech12], [Mertens et al.12, Shao et al.14, Cuevas-Maraver et al.14]
Assuming linear stability, one tries to prove asymptotic stability • [Pelinovsky & Stefanov12][Boussaid & Cuccagna12][Comech, Phan, Stefanov14]
4 Nonrelativistic limit of NLD: ω . m
[Ounaies00, Guan08] v(x) Solitary wave: ψ(x,t) = e−iωt; v, u C2 x R3 u(x) ∈ ∈ 0 σ 1 0 iψ˙ = i ·∇ + m (ψψ¯ )k ψ, − σ 0 − 0 1 n ·∇ − o v u v ω iσ + m v 2k u ≈ − ·∇ v −| | u − If ω . m, u v 1: 2mu iσ v, v satisfies NLS: | |≪| | ≪ ≈ − ·∇ 1 (m ω)v = ∆v v 2kv − − −2m −| |
Scaling: v(x) = ǫ1/kΦ(ǫx), ǫ = √m ω, − 1 Φ = ∆Φ Φ 2kΦ, Φ(x) R, x Rn. − −2m −| | ∈ ∈
5 NLD: linearization at a solitary wave
Given φ (x)e−iωt, ω ( m,m), consider ψ(x,t) = φ (x)+ r(x,t) e−iωt ω ∈ − ω Linearized eqn on r(x,t), i∂ r = D r ωr + ... t m −
(m + ω)i Re r 0 Dm ω + ... Re r ∂t = − Im r Dm + ω + ... 0 Im r − Aω | {z } (m − ω)i ∂tR = AωR; σ(Aω) iR (“spectral stability”) ??? ⊂ ✲
σess(Aω) σ(D ω) m − r0 m ω m ω − − −
6 σ(Aω)
Linear instability of NLD 2mi (m + ω)i i∂ ψ = D ψ (ψψ¯ )kβψ, x Rn t m − ∈
−iωt ψ(x,t) = φω(x)e (m − ω)i Theorem 1 ([Comech, Guan, Gustafson12]). t tt ✲ ✲ −λω λω If k > 2/n, so that NLSk is linearly unstable, then for ω . m, λ σ (A ), ∃ ± ω ∈ d ω Re λω > 0, λω 0 ω−→→m
Unstable for n = 1, k > 2 (above quintic) −2mi Unstable for n = 2, k > 1 (above cubic) ω < m ω m Unstable for n = 3, k > 2/3 = Proof: Rescale; use Schur reduction and Rayleigh-Schrodinger¨ perturbation theory. ✷
7 σ(Aω) Bifurcations from σess (m + ω0)i
λω Let 0 ω0 m λ 0 ω r r r r✈r ≤ ≤ r r r r Theorem 2 ([Boussaid & Comech16]). (m − ω0)i Assume: λω σp(Aω), Re λω = 0, ∈ 6 t ✲ R λω λω0 i ω−→→ω0 ∈ Then: r r r r r r r✈r r λ m + ω0, | ω0 | ≤ λ σ (A ) 0 i(m + ω0) ω0 ∈ p ω0 ∪ { } ∪ { }
LAP methods of [Agmon75], [Berthier & Georgescu87]:
u 1 c (D z)u 2 , z C [ m,m], s> 1/2, z m > δ k kH−s ≤ z,s,δk m − kLs ∈ \ − | ± |
s u 1 := (1 + x ) u) 1 k kHs k | | kH
8 σ(Aω)
Linear stability of NLD 2mi (m + ω)i 2ωi t i∂ ψ = D ψ (ψψ¯ )kβψ, x Rn t m − ∈
−iωt ψ(x,t) = φω(x)e (m − ω)i t ✲ ✲ t Theorem 3 ([Comech11]). If n 3, then λ = 2ωi σ (A ). ≤ ± ∈ p ω
−2ωi t Theorem 4 ([Boussaid & Comech16]). n 3, k . 2/n: −2mi ≤ −iωt φωe are linearly stable for ω . m ω < m ω = m Including the “charge-critical” case k = 2/n
9 Eigenvalues λ = 2ωi in the Soler model ±
λ = 2ωi is bad for proving asymptotic stability! ±
[Boussaid & Cuccagna12], [Comech, Phan, Stefanov14] Asymptotic stability of solitons in Soler model for “radially symmetric” case:
1. perturbations orthogonal to translations; 2. perturbations orthogonal to 2ωi eigenvectors ±
10 SU(1,1) invariance of the Soler model and the Soler charge
Theorem 5 ([Boussaid & Comech16]). 1. Soler model hamiltonian is invariant under transformations ψ(x,t) (a ibγ2C)ψ(x,t), a, b C, a 2 b 2 = 1 −→ − ∈ | | −| | a b 2. These transformations form the group SU(1, 1) U(1) ⊃ ¯b a¯
3. Conserved charges: Q = ψ∗ψdx > 0 Λ = R ψt( iγ2)ψ dx C − ∈ R 4. Bi-frequency solutions: if φ(x)e−iωt is a solitary wave, so is ψ(x,t) = ae−iωtφ(x) beiωtiγ2C(φ(x)) −
Note: if b 1, then ψ(x,t) e−iωt φ(x) be2iωtiγ2C(φ(x)) | | ≪ ≈ −
11 Q(ω), 10 10
8 8 E(ω)
6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1 1
0.8 0.8
0.6 0.6
0.4 0.4
λ iR+ ∈ 0.2 0.2
0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Figure 1: (1+1)D Soler model. LEFT: k = 2 (“charge-critical”); RIGHT: k = 3. TOP ROW: charge and energy of the solitary waves as functions of ω ∈ (0, 1). BOTTOM ROW: Spectrum on the upper half of the imaginary axis. Note the exact eigenvalue λ = 2ωi.
12 1
0.5 ) λ 0 Im(
−0.5
−1 0 0.5 1 ω
Figure 2: (2+1)D cubic “charge-critical” Soler model [Cuevas-Maraver et al.16]
13 1
0.5 ) λ 0 Im(
−0.5
−1 0 0.5 1 ω
Figure 3: (3+1)D cubic “charge-supercritical” Soler model [Cuevas-Maraver et al.16]
14 Theorem 6 ([Berkolaiko, Comech, Sukhtayev15]). Both Q′(ω) = 0 and E(ω) = 0 indicate collision of eigenvalues at λ = 0
15
−iωt 10 Q(φωe )
5 −iωt E(φωe ) 0
-5 -1 -0.5 0 0.5 1
1
0.8
0.6
λ iR+ 0.4 ∈ 0.2
0 -1 -0.5 0 0.5 1
m<ω 15 References [Abenda98] S. Abenda, Solitary waves for Maxwell-Dirac and Coulomb-Dirac models, Ann. Inst. H. Poincare´ Phys. Theor.,´ 68 (1998), pp. 229–244. [Agmon75] S. Agmon, Spectral properties of Schrodinger¨ operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), pp. 151–218. [Alvarez & Carreras81] A. Alvarez & B. Carreras, Interaction dynamics for the solitary waves of a nonlinear Dirac model, Phys. Lett. A, 86 (1981), pp. 327–332. [Alvarez & Soler83] A. Alvarez & M. Soler, Energetic stability criterion for a nonlinear spinorial model, Phys. Rev. Lett., 50 (1983), pp. 1230–1233. [Alvarez & Soler86] A. Alvarez & M. Soler, Stability of the minimum solitary wave of a nonlinear spinorial model, Phys. Rev. D, 34 (1986), pp. 644–645. [Berkolaiko & Comech12] G. Berkolaiko & A. Comech, On spectral stability of solitary waves of nonlinear Dirac equation in 1D, Math. Model. Nat. Phenom., 7 (2012), pp. 13–31. 16 6 Q(ω) 5 4 3 2 E(ω) 1 0 1 -1 -0.5 0 0.5 1 0.8 0.6 0.4 λ iR+ ∈ 0.2 0 -1 -0.5 0 0.5 1 m<ω 4 3 2 1 E(ω) 0 -11 -1 -0.5 0 0.5 1 0.8 0.6 0.4 λ iR+ ∈ 0.2 0 -1 -0.5 0 0.5 1 Figure 6: Massive Thirring model with k = 2, “charge critical”. TOP: energy and charge as functions of ω ∈ (−1, 1). BOTTOM: The spectrum of the linearization at a solitary wave. 18 3 Q(ω) 2 1 0 E(ω) -11 -1 -0.5 0 0.5 1 0.8 0.6 0.4 λ iR+ ∈ 0.2 0 -1 -0.5 0 0.5 1 Figure 7: Massive Thirring model with k = 3. TOP: energy and charge as functions of ω ∈ (−1, 1). BOTTOM: The spectrum of the linearization at a solitary wave. 19 [Berkolaiko et al.15] G. Berkolaiko, A. Comech, & A. Sukhtayev, Vakhitov-Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classi- cal self-interacting spinor fields, Nonlinearity, 28 (2015), pp. 577–592. [Berthier & Georgescu87] A. Berthier & V. Georgescu, On the point spectrum of Dirac op- erators, J. Funct. Anal., 71 (1987), pp. 309–338. [Bogolubsky79] I. L. Bogolubsky, On spinor soliton stability, Phys. Lett. A, 73 (1979), pp. 87–90. [Boussaid & Comech16] N. Boussaid & A. Comech, On spectral stability of nonlinear Dirac equation, J. Funct. Anal., (2016). [Boussaid & Cuccagna12] N. Boussaid & S. Cuccagna, On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations, 37 (2012), pp. 1001– 1056. [Cazenave & Vazquez´ 86] T. Cazenave & L. Vazquez,´ Existence of localized solutions for a classical nonlinear Dirac field, Comm. Math. Phys., 105 (1986), pp. 35–47. [Comech11] A. Comech, On the meaning of the Vakhitov-Kolokolov stability criterion for the nonlinear Dirac equation, ArXiv e-prints, (2011). 20 [Cooper et al.10] F. Cooper, A. Khare, B. Mihaila, & A. Saxena, Solitary waves in the non- linear Dirac equation with arbitrary nonlinearity, Phys. Rev. E, 82 (2010), p. 036604. [Cuevas-Maraver et al.14] J. Cuevas-Maraver, P. G. Kevrekidis, & A. Saxena, Solitary waves in a discrete nonlinear Dirac equation, ArXiv e-prints, (2014). [Cuevas-Maraver et al.16] J. Cuevas-Maraver, P. G. Kevrekidis, A. Saxena, A. Comech, & R. Lan, Stability of solitary waves and vortices in a 2D nonlinear Dirac model, Phys. Rev. Lett., 116 (2016), p. 214101. [Dirac28] P. Dirac, The quantum theory of the electron, Proc. R. Soc. A, 117 (1928), pp. 610–624. [Esteban et al.96] M. J. Esteban, V. Georgiev, & E.´ Ser´ e,´ Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations, Calc. Var. Partial Differential Equations, 4 (1996), pp. 265–281. [Finkelstein et al.56] R. Finkelstein, C. Fronsdal, & P. Kaus, Nonlinear spinor field, Phys. Rev., 103 (1956), pp. 1571–1579. [Finkelstein et al.51] R. Finkelstein, R. LeLevier, & M. Ruderman, Nonlinear spinor fields, Phys. Rev., 83 (1951), pp. 326–332. 21 [Gross & Neveu74] D. J. Gross & A. Neveu, Dynamical symmetry breaking in asymptoti- cally free field theories, Phys. Rev. D, 10 (1974), pp. 3235–3253. [Guan08] M. Guan, Solitary wave solutions for the nonlinear Dirac equations, ArXiv e- prints, (2008). [Heisenberg57] W. Heisenberg, Quantum theory of fields and elementary particles, Rev. Mod. Phys., 29 (1957), pp. 269–278. [Ivanenko38] D. D. Ivanenko, Notes to the theory of interaction via particles, Zh. Eksp.´ Teor. Fiz, 8 (1938), pp. 260–266. [Mertens et al.12] F. G. Mertens, N. R. Quintero, F. Cooper, A. Khare, & A. Saxena, Nonlin- ear Dirac equation solitary waves in external fields, Phys. Rev. E, 86 (2012), p. 046602. [Ounaies00] H. Ounaies, Perturbation method for a class of nonlinear Dirac equations, Dif- ferential Integral Equations, 13 (2000), pp. 707–720. [Pelinovsky & Stefanov12] D. E. Pelinovsky & A. Stefanov, Asymptotic stability of small gap solitons in nonlinear Dirac equations, J. Math. Phys., 53 (2012), pp. 073705, 27. [Schrodinger¨ 26] E. Schrodinger,¨ Quantisierung als Eigenwertproblem, Ann. Phys., 386 (1926), pp. 109–139. 22 [Shao et al.14] S. Shao, N. R. Quintero, F. G. Mertens, F. Cooper, A. Khare, & A. Saxena, Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity, ArXiv e-prints, (2014). [Soler70] M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1 (1970), pp. 2766–2769. [Strauss & Vazquez´ 86] W. A. Strauss & L. Vazquez,´ Stability under dilations of nonlinear spinor fields, Phys. Rev. D (3), 34 (1986), pp. 641–643. [Thirring58] W. E. Thirring, A soluble relativistic field theory, Ann. Physics, 3 (1958), pp. 91–112. 23