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Linear stability and instability of self-interacting spinor field A - - PowerPoint PPT Presentation

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Linear stability and instability of self-interacting spinor field A NDREW C OMECH Texas A&M University, College Station, TX, USA October, 2013 Einstein: E 2 = p 2 + m 2 , odinger: ( i t ) 2 = ( i ) 2 + m 2 Schr p 2

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Linear stability and instability of self-interacting spinor field

ANDREW COMECH

Texas A&M University, College Station, TX, USA October, 2013

Einstein: E2 = p2 + m2, Schr¨

  • dinger: (i∂t)2ψ = (−i∇)2ψ + m2ψ

E =

  • m2 + p2 ≈ m +

p2 2m,

[Schr¨

  • dinger26]: i∂tψ =

1 2m(−i∇)2ψ

[Dirac28]: E =

  • p2 + m2 = α · p + βm,

i∂tψ = (−iα · ∇ + βm)

  • Dm

ψ, ψ(x, t)∈C4, x∈R3 αj (1≤j≤3) and β are self-adjoint and such that D2

m = −∆ + m2

Standard choice: αj =

  • σj

σj

  • (Pauli matrices), β =
  • I2

−I2

  • 1
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SLIDE 2

Self-interacting spinors

Models of self-interacting spinor field: [Ivanenko38, Finkelstein et al.51, Finkelstein et al.56, Heisenberg57] [...] Massive Thirring model [Thirring58] in (n+1)D: LMTM = ¯ ψ(iγµ∂µ − m)ψ + ¯ ψγµψ ¯ ψγµψ k+1

2

k > 0 (V-V) Soler model [Soler70] in (n+1)D: LSoler = ¯ ψ(iγµ∂µ − m)ψ + ( ¯ ψψ)k+1 (S-S) In ( 1+1 )D: massive Gross-Neveu model [Gross & Neveu74, Lee & Gavrielides75] 2

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Soler model: NLD with scalar-scalar self-interaction

i∂tψ = (−iα · ∇ + mβ)

  • Dm

ψ − ( ¯ ψψ)kβψ, ψ ∈ CN, x ∈ Rn

  • [Soler70, Cazenave & V´

azquez86]: existence of solitary waves in R3, ψ(x, t) = φω(x)e−iωt, ω ∈ (0, m)

  • Attempts at stability: [Bogolubsky79, Alvarez & Soler86, Strauss & V´

azquez86] ...

  • Numerics [Alvarez & Carreras81, Alvarez & Soler83, Berkolaiko & Comech12] suggest that

(all?) solitary waves in 1D cubic Soler model are stable

  • Assuming linear stability, one tries to prove asymptotic stability

[Pelinovsky & Stefanov12, Boussaid & Cuccagna12] 3

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Nonrelativistic limit of NLD: ω → m

Solitary wave: ψ(x, t) = φ(x) ρ(x)

  • e−iωt;

φ, ρ ∈ C2 i ˙ ψ =

  • − i
  • σ · ∇

σ · ∇

  • + mβ
  • ψ − ( ¯

ψψ)kβψ, ω φ ρ

  • ≈ −iσ · ∇

ρ φ

  • + m

φ −ρ

  • − |φ|2k

φ −ρ

  • If ω m:

2mρ ≈ −iσ · ∇φ, φ satisfies NLS: −(m − ω)φ = − 1 2m∆φ − |φ|2kφ. Scaling: φ(x) = ǫ1/kΦ(ǫx), ǫ = √m − ω −Φ = − 1 2m∆Φ − |Φ|2kΦ 4

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NLD: linearization at a solitary wave

Given φω(x)e−iωt, consider ψ(x, t) =

  • φω(x) + r(x, t)
  • e−iωt

Linearized eqn on r(x, t) ∈ CN, i∂tr = Dmr − ωr + . . . ∂t Re r Im r

  • =
  • Dm − ω + ...

−Dm + ω + ...

  • Aω

Re r Im r

  • (m + ω)i

(m − ω)i

✲ σess(Aω) m − ω −m − ω σ(Dm − ω) r 5

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Linear instability of NLD

ω = m 2mi −2mi

✲

ω < m (m + ω)i (m − ω)i λω −λω

t t t ✲ σ(Aω) Theorem 1 ([Comech & Guan & Gustafson12]). If NLSk is linearly unstable, then for ω m, ∃ ±λω ∈ σd(Aω), Re λω > 0, λω − →

ω→m 0

1D, above quintic 2D, above cubic 3D cubic and above Proof: Rescale; use Rayleigh-Schr¨

  • dinger perturbation theory.

✷ 6

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Linear stability of NLD

ω = m 2mi −2mi

✲

ω < m (m + ω)i 2ωi (m − ω)i λω −λω

t t t t t ✲ σ(Aω) Theorem 2 ([Boussaid & Comech12]). Assume λω ∈ σp(Aω), ω m

  • 1. λω −

→

ω→m {0 ; ± 2mi}.

  • 2. If Re λω = 0, then λω −

→

ω→m 0,

Λ := lim

ω→m

λω (m − ω) ∈ σp(NLSk)

  • 3. Λ = 0 unless critical case:

2D quintic; 3D cubic Proof: Limiting absorption principle [Agmon75, Berthier & Georgescu87] ✷ 7

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SLIDE 8

Linear stability of NLD

ω = m 2mi −2mi

✲

ω < m (m + ω)i (m − ω)i

t t t ✲ σ(Aω) Corollary 3 ([Boussaid & Comech12]). 1D cubic: φωe−iωt are linearly stable for ω m Remark 4. Also true for 1D cubic and 2D quintic (“charge-critical NLS”) 8

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0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

  • 1
  • 0.5

0.5 1

Figure 1: Upper half of the spectral gap. TOP: 1D cubic Soler BOTTOM: 1D cubic massive Thirring 9

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0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

  • 1
  • 0.5

0.5 1

Figure 2: 1D quintic (“charge critical”). TOP: Soler; BOTTOM: massive Thirring 10

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0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

  • 1
  • 0.5

0.5 1

Figure 3: 1D, seventh order. Soler and MTM 11

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SLIDE 12

Bifurcations from σess

r r r r r r r r r r r r r r r r r r r r

(m + Ω)i λΩ λω (m − Ω)i

t ✈ ✈ ✲ σ(AΩ) Let Ω ∈ (0, m) Theorem 5 ([Boussaid & Comech12]). If λω ∈σp(Aω), Re λω = 0, λω − →

ω→Ω λΩ ∈ iR

then λΩ ∈ σp(AΩ), |λΩ| ≤ m + Ω Moreover, λ ∈ σp ∩ iR ⇒ |λ| ≤ m + |Ω| 12

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Theorem 6 ( [[Berkolaiko & Comech & Sukhtyaev13]). Q′(ω) = 0 and E(ω) = 0 correspond to the boundary of the linear instability region

−5 5 10 15 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Figure 4: quadratic MTM. TOP: Charge (······) and energy (− −) as functions of ω. BOTTOM: Purely imaginary eigenvalues (•, ) of the linearized equation in the spectral gap. 13

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SLIDE 14

References

[Agmon75] S. Agmon, Spectral properties of Schr¨

  • dinger 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. [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. 14

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[Boussaid & Comech12] N. Boussaid & A. Comech, On spectral stability of nonlinear Dirac equation, ArXiv e-prints, (2012). [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 & V´ azquez86] T. Cazenave & L. V´ azquez, Existence of localized solutions for a classical nonlinear Dirac field, Comm. Math. Phys., 105 (1986), pp. 35–47. [Dirac28] P. Dirac, The quantum theory of the electron, Proc. R. Soc. A, 117 (1928), pp. 610–624. [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.

[Gross & Neveu74] D. J. Gross & A. Neveu, Dynamical symmetry breaking in asymptoti- cally free field theories, Phys. Rev. D, 10 (1974), pp. 3235–3253. [Heisenberg57] W. Heisenberg, Quantum theory of fields and elementary particles, Rev.

  • Mod. Phys., 29 (1957), pp. 269–278.

15

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[Ivanenko38] D. D. Ivanenko, Notes to the theory of interaction via particles, Zh. ´

  • Eksp. Teor.

Fiz, 8 (1938), pp. 260–266. [Lee & Gavrielides75] S. Y. Lee & A. Gavrielides, Quantization of the localized solutions in two-dimensional field theories of massive fermions, Phys. Rev. D, 12 (1975), pp. 3880–3886. [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. [Schr¨

  • dinger26] E. Schr¨
  • dinger, Quantisierung als Eigenwertproblem, Ann. Phys., 386

(1926), pp. 109–139. [Soler70] M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys.

  • Rev. D, 1 (1970), pp. 2766–2769.

[Strauss & V´ azquez86] W. A. Strauss & L. V´ azquez, 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. 16