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<m − Figure 4: “Quadratic” massive Thirring model, k = 1/2. TOP: energy and charge as functions of ω ∈ (−m, m). BOTTOM: The spectrum of the linearization at a solitary wave. Dotted eigenvalue collides with its opposite at the origin when ω⋆ ≈ −0.6276m, where E(ω⋆) = 0 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<m − Figure 5: Massive Thirring model with k = 1. TOP: energy and charge as functions of ω ∈ (−1, 1). BOTTOM: The spectrum of the linearization at a solitary wave on the upper half of the imaginary axis. No nonzero eigenvalues in completely integrable model. 17 5 Q(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).