We consider a ground state (soliton) of NLS in dimension three. We prove
 that if the soliton is orbitally stable, then it is also

asymptotically stable. The main assumptions are transversal
 nondegeneracy of the manifold of the ground states, linear dispersion
 (in the form of Strichartz estimates) and nonlinear Fermi Golden

Rule

D. Bambusi

We consider a ground state (soliton) of NLS in dimension three. We prove
 that if the soliton is orbitally stable, then it is also

asymptotically stable. The main assumptions are transversal
 nondegeneracy of the manifold of the ground states, linear dispersion
 (in the form of Strichartz estimates) and nonlinear Fermi Golden

Rule

AIR Universita degli studi di Milano

The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryAsymptotic Stability of Solitons.D. Bambusi11Dipartimento di Matematica “F.Enriques” - Universita` degli studi di MilanoTine´e, February 8, 2011The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryOutline1 The Problem2 A model problem3 Recall of BC094 Connection with NLS5 Marsden-Weinstein reduction6 Darboux theorem7 Perturbation theoryThe Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryNLSModel problemiψt = −∆ψ − β′(|ψ|2)ψ , x ∈ R3 ,|β(u)(k)| ≤ Ck〈x〉p˜−k , p˜ ≤ 1 + 2d − 2 , d = 3Symmetries:translations (ψ, qi ) 7→ ψ(.− qiei ), generated by −i∂xjGauge (ψ, q4) 7→ eiq4ψ, generated by i.Conservation laws:P j(ψ) :=∫ψ∂j ψ¯ − ψ¯∂jψ2P4(ψ) :=∫|ψ|2The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryGround StatesLook for special solutionsψ(x , t) = e−i(ω4t+q4)ηp(x − (ωj t + qj)ej)ηp is a critical point ofH :=∫|∇ψ|2 − β(|ψ|2)restricted toSp :={ψ : P j(ψ) = pj , j = 1, ..., 4}Ground state. A ground state is the minimum.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryTheoremAssumeAssumptions on the linearized operator.Fermi golden rule (probably generic generic: work in progress)infp,q∥∥ψ0 − e−iq4ηp(.− qiej)∥∥H1  1TheoremThere exist functions ω(t), p(t) q(t) having a limit as t → +∞ and astate ψ∞ such that, writingψ(x , t) = e−iy4(t))ηp(t)(x − y(t)) + χ(x , t) ,yj(t) = ωj(t)t + qj(t) , |qj(t)|  1 , j = 1, ...., 4one haslimt→+∞∥∥χ(t)− e it∆ψ∞∥∥L6 = 0 .The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryCommentsWhat’s new? Old results when the Floquet spectrum has at most 1eigenvalue: Weinstein, Soffer-Weinstein, Buslaev-Perelman,Cuccagna, Perelman.Ideas from D.B.-Cuccagna (on Klein Gordon), Cuccagna (case withpotential), Perelman (no eigenvalues, energy space).Key difficulty: the generators of the symmetries are unbounded.Development of reduction theory, Darboux theory and Normal formtheory with only continuous transformations.Validity of Strichartz estimates for the relevant operators.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryCommentsWhat’s new? Old results when the Floquet spectrum has at most 1eigenvalue: Weinstein, Soffer-Weinstein, Buslaev-Perelman,Cuccagna, Perelman.Ideas from D.B.-Cuccagna (on Klein Gordon), Cuccagna (case withpotential), Perelman (no eigenvalues, energy space).Key difficulty: the generators of the symmetries are unbounded.Development of reduction theory, Darboux theory and Normal formtheory with only continuous transformations.Validity of Strichartz estimates for the relevant operators.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryCommentsWhat’s new? Old results when the Floquet spectrum has at most 1eigenvalue: Weinstein, Soffer-Weinstein, Buslaev-Perelman,Cuccagna, Perelman.Ideas from D.B.-Cuccagna (on Klein Gordon), Cuccagna (case withpotential), Perelman (no eigenvalues, energy space).Key difficulty: the generators of the symmetries are unbounded.Development of reduction theory, Darboux theory and Normal formtheory with only continuous transformations.Validity of Strichartz estimates for the relevant operators.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryOutline1 The Problem2 A model problem3 Recall of BC094 Connection with NLS5 Marsden-Weinstein reduction6 Darboux theorem7 Perturbation theoryThe Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryModel probelmThe equation: iψt = −∆ψ + Vψ + δHPδψ¯(x) .H0 := −∆ + V with one eigenvalue H0e = ωeSpectral coordinates ψ = ξe + f :H0 = 〈f¯ ;Bf 〉+ ω|ξ|2Model nonlinearityHP := ξ¯ν〈Φ¯; f 〉+ ξν〈Φ; f¯ 〉Equationsξ˙ = −iωξ − iνξ¯ν〈Φ¯; f 〉f˙ = −i(Bf + ξνΦ)Further decoupling g = f + ξνΨ: if Ψ is such that(B − νω)Ψ = Φthen g˙ = iBg + O(|ξ|ν |f |+ |ξ|2ν−1)The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryDissipationDefineR∓ν := lim→0+(B − ων ± i)−1 , Ψ = R−ν Φthen Ψ(x) ∼ 〈x〉−1. Plug in the equation for ξ:ξ˙ = −iωξ − i |ξ|2ν−1 〈Φ¯;R+ν Φ〉ξ + O(ξν−1|〈Φ; g〉|)Plemelji formula:R−ν ≡ (B − νω − i0)−1 = PV (B − νω)− ipiδ(B − νω) .implies 〈Φ¯;R−ν Φ〉 = a− ib, b ≥ 0ξ˙ = −iωξ − ia|ξ|2ν−1ξ − b|ξ|2ν−1ξ+h.o.t.ddt|ξ|2 = −2b|ξ|2ν =⇒ |ξ|ν ∈ L2tUse normal form to reduce to the model problem.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryDissipationDefineR∓ν := lim→0+(B − ων ± i)−1 , Ψ = R−ν Φthen Ψ(x) ∼ 〈x〉−1. Plug in the equation for ξ:ξ˙ = −iωξ − i |ξ|2ν−1 〈Φ¯;R+ν Φ〉ξ + O(ξν−1|〈Φ; g〉|)Plemelji formula:R−ν ≡ (B − νω − i0)−1 = PV (B − νω)− ipiδ(B − νω) .implies 〈Φ¯;R−ν Φ〉 = a− ib, b ≥ 0ξ˙ = −iωξ − ia|ξ|2ν−1ξ − b|ξ|2ν−1ξ+h.o.t.ddt|ξ|2 = −2b|ξ|2ν =⇒ |ξ|ν ∈ L2tUse normal form to reduce to the model problem.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryDissipationDefineR∓ν := lim→0+(B − ων ± i)−1 , Ψ = R−ν Φthen Ψ(x) ∼ 〈x〉−1. Plug in the equation for ξ:ξ˙ = −iωξ − i |ξ|2ν−1 〈Φ¯;R+ν Φ〉ξ + O(ξν−1|〈Φ; g〉|)Plemelji formula:R−ν ≡ (B − νω − i0)−1 = PV (B − νω)− ipiδ(B − νω) .implies 〈Φ¯;R−ν Φ〉 = a− ib, b ≥ 0ξ˙ = −iωξ − ia|ξ|2ν−1ξ − b|ξ|2ν−1ξ+h.o.t.ddt|ξ|2 = −2b|ξ|2ν =⇒ |ξ|ν ∈ L2tUse normal form to reduce to the model problem.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryOutline1 The Problem2 A model problem3 Recall of BC094 Connection with NLS5 Marsden-Weinstein reduction6 Darboux theorem7 Perturbation theoryThe Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryOn the wave equationThe equation.utt −∆u + Vu + m2u = −β′(u) , (t, x) ∈ R× R3with −∆ + V (x) + m2 a positive Schro¨dinger operator (V smoothand fast decaying),β′ a smooth function function fulfilling |β(u)| ≤ Cu4Consequence. −∆ + V has finitely many eigenvalues−λ21 ≤ · · · ≤ −λ2n ≤ 0and σc(−∆ + V ) = [0,+∞)The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryThe theorem of BC09Denote K0(t) :=sin(t√−∆ + m2)√−∆ + m2Remark u(t) := K ′0(t)u0 + K0(t)v0 solvesutt −∆u + m2u = 0u(x , 0) = u0(x) u˙(x , 0) = v0(x) .TheoremThere exists 0 > 0 such that, if‖(u0, v0)‖H1×L2 < 0then there exist (u±, v±) ∈ H1 × L2 such thatlimt→±∞ ‖u(t)− K′0(t)u± + K0(t)v±‖H1 = 0The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryOutline1 The Problem2 A model problem3 Recall of BC094 Connection with NLS5 Marsden-Weinstein reduction6 Darboux theorem7 Perturbation theoryThe Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryMarsden-Weinstein reduction - generel factsConnection between the two problems: Marsden-Weinstein reduction.Symplectic manifold: (M, ω), J Poisson tensorSymmetry group: (q, u) 7→ eqJAu, generated byP(u) := 12〈u;Au〉 ,Invariant Hamiltonian: H, s.t. H(u) = H(eqJAu),Reduced system: Sp := {u ∈M : P(u) = p} andMp := Sp/ ' ,(u ∼ u′ ⇐⇒ u′ = eqJAu)Explicit construction: see the blackboard!Ω := i∗ω, and Hr := i∗H = H ◦ i .Ground state, ηp: minimum of Hr !The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryMarsden-Weinstein reduction - generel factsConnection between the two problems: Marsden-Weinstein reduction.Symplectic manifold: (M, ω), J Poisson tensorSymmetry group: (q, u) 7→ eqJAu, generated byP(u) := 12〈u;Au〉 ,Invariant Hamiltonian: H, s.t. H(u) = H(eqJAu),Reduced system: Sp := {u ∈M : P(u) = p} andMp := Sp/ ' ,(u ∼ u′ ⇐⇒ u′ = eqJAu)Explicit construction: see the blackboard!Ω := i∗ω, and Hr := i∗H = H ◦ i .Ground state, ηp: minimum of Hr !The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryMarsden-Weinstein reduction - generel factsConnection between the two problems: Marsden-Weinstein reduction.Symplectic manifold: (M, ω), J Poisson tensorSymmetry group: (q, u) 7→ eqJAu, generated byP(u) := 12〈u;Au〉 ,Invariant Hamiltonian: H, s.t. H(u) = H(eqJAu),Reduced system: Sp := {u ∈M : P(u) = p} andMp := Sp/ ' ,(u ∼ u′ ⇐⇒ u′ = eqJAu)Explicit construction: see the blackboard!Ω := i∗ω, and Hr := i∗H = H ◦ i .Ground state, ηp: minimum of Hr !The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryDifficultyOnly continuous group actions:u(.) 7→ u(.− q), generated by ∂tu = −∂xu.(1) Does reduction theory holds, and in particularIs the reduced manifold a manifold?Can one define the reduced system?(2) Canonical coordinates are needed: is it possible to prove Darbouxtheorem?(3) develop transformation theory with unbounded generators(4) Dispersive estimates: do Strichartz estimates persist underunbounded perturbation?The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryOutline1 The Problem2 A model problem3 Recall of BC094 Connection with NLS5 Marsden-Weinstein reduction6 Darboux theorem7 Perturbation theoryThe Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryFrameworkPhase space:Hk , k ∈ Z Scale of Hilbert spaces, 〈.; .〉 scalar prod of H0J : Hk 7→ Hk−d Poisson tensorE := J−1, ω(U1,U2) := 〈EU1;U2〉XH := J∇H Hamiltonian vector fieldThe system: H :=12〈A0u; u〉+ HP(u); A0 : Hk → Hk−d0Symmetries: P j(u) := 12〈Aju; u〉;Aj : Hk → Hk−dj , JAj generate a flow: eqJAj : Hk 7→ Hk , ∀k.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryGround state and decomposition of the spaceA0ηp +∇HP(ηp)−∑jλjAjηp = 0Assumptions:(1) Rn ⊃ I 3 p 7→ ηp ∈ H∞ is smooth.Normalization condition P j(ηp) = pj .(2)Sp∈I {ηp} is isotropicKey remark: q 7→ eqJAjηp is smooth!Consequence: T :=⋃q,pePqjJAjηp ' I × (Tl × Rn−l),Natural decomposition: H0 ≡ TηpH0 ' TηpT ⊕ TωηpTwith TωηpT :={U : ω(U;X ) = 0 , ∀X ∈ TηpT}.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryGround state and decomposition of the spaceA0ηp +∇HP(ηp)−∑jλjAjηp = 0Assumptions:(1) Rn ⊃ I 3 p 7→ ηp ∈ H∞ is smooth.Normalization condition P j(ηp) = pj .(2)Sp∈I {ηp} is isotropicKey remark: q 7→ eqJAjηp is smooth!Consequence: T :=⋃q,pePqjJAjηp ' I × (Tl × Rn−l),Natural decomposition: H0 ≡ TηpH0 ' TηpT ⊕ TωηpTwith TωηpT :={U : ω(U;X ) = 0 , ∀X ∈ TηpT}.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryGround state and decomposition of the spaceA0ηp +∇HP(ηp)−∑jλjAjηp = 0Assumptions:(1) Rn ⊃ I 3 p 7→ ηp ∈ H∞ is smooth.Normalization condition P j(ηp) = pj .(2)Sp∈I {ηp} is isotropicKey remark: q 7→ eqJAjηp is smooth!Consequence: T :=⋃q,pePqjJAjηp ' I × (Tl × Rn−l),Natural decomposition: H0 ≡ TηpH0 ' TηpT ⊕ TωηpTwith TωηpT :={U : ω(U;X ) = 0 , ∀X ∈ TηpT}.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryGround state and decomposition of the spaceA0ηp +∇HP(ηp)−∑jλjAjηp = 0Assumptions:(1) Rn ⊃ I 3 p 7→ ηp ∈ H∞ is smooth.Normalization condition P j(ηp) = pj .(2)Sp∈I {ηp} is isotropicKey remark: q 7→ eqJAjηp is smooth!Consequence: T :=⋃q,pePqjJAjηp ' I × (Tl × Rn−l),Natural decomposition: H0 ≡ TηpH0 ' TηpT ⊕ TωηpTwith TωηpT :={U : ω(U;X ) = 0 , ∀X ∈ TηpT}.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryA nonlinear constructionLevel surface of P: Sp :={u : P j(u) = pj , ∀j} 3 ηpInvariance: ePqjJAjηp ⊂ Sp.Transversality: if Mp ⊂ Sp is such that TηpMp = TωηpT then itshould be a eqivalent (locally) to the reduced system.Explicit construction of the reduced systemFix p0 ∈ I .Local model: V := TωηpT , Vk := Hk ∩ V, (k ≥ 0), V−k dual of Vk .Look for Vk 3 φ 7→ p(φ) ∈ I s.t.u := ηp(φ) + Πp(φ)φ ∈ Sp0 .The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryA nonlinear constructionLevel surface of P: Sp :={u : P j(u) = pj , ∀j} 3 ηpInvariance: ePqjJAjηp ⊂ Sp.Transversality: if Mp ⊂ Sp is such that TηpMp = TωηpT then itshould be a eqivalent (locally) to the reduced system.Explicit construction of the reduced systemFix p0 ∈ I .Local model: V := TωηpT , Vk := Hk ∩ V, (k ≥ 0), V−k dual of Vk .Look for Vk 3 φ 7→ p(φ) ∈ I s.t.u := ηp(φ) + Πp(φ)φ ∈ Sp0 .The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryReductionDefine: i(φ) byVk 3 φ 7→ ηp(P(φ),φ) + Πp(P(φ),φ)φ =: i(φ) ∈ Sp0Hr := i∗H , Ω := i∗ω .Vector field: Hr defines a Hamiltonian vector field in V.AssumptionsXH defines a local flow on Hk0 which leaves invariant Hk ∀k ≥ k0.The same for XHr .Theorem∃C > 0 s.t., ifdk0 (u0, T ) < C , u0 ∈ Sp0 .then ∃q(t) s.t. u(t) = eqj (t)JAj i(φ(t)).The converse is also true.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryReductionDefine: i(φ) byVk 3 φ 7→ ηp(P(φ),φ) + Πp(P(φ),φ)φ =: i(φ) ∈ Sp0Hr := i∗H , Ω := i∗ω .Vector field: Hr defines a Hamiltonian vector field in V.AssumptionsXH defines a local flow on Hk0 which leaves invariant Hk ∀k ≥ k0.The same for XHr .Theorem∃C > 0 s.t., ifdk0 (u0, T ) < C , u0 ∈ Sp0 .then ∃q(t) s.t. u(t) = eqj (t)JAj i(φ(t)).The converse is also true.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryOutline1 The Problem2 A model problem3 Recall of BC094 Connection with NLS5 Marsden-Weinstein reduction6 Darboux theorem7 Perturbation theoryThe Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theorySearch for canonical coordinatesΩφ(Φ; Φ) = 〈E(φ)Φ; Φ〉 , E(φ) = E + O(φ2) .Darboux theoremThere exists a map of the formφ = F(φ′) = ePj qjJAj(φ′ + S(N, φ′)) , N j := P j(φ′) , (1)where the following following properties hold1. qi (N, φ) is defined on Rn × V−∞2. S : Rn × V−k 7→ S(N, φ) ∈ V l is smoothing.3. in terms of the variables φ′ the symplectic form is given byΩ(Φ′1; Φ′2) = 〈EΦ′1; Φ′2〉 . (2)The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theorySearch for canonical coordinatesΩφ(Φ; Φ) = 〈E(φ)Φ; Φ〉 , E(φ) = E + O(φ2) .Darboux theoremThere exists a map of the formφ = F(φ′) = ePj qjJAj(φ′ + S(N, φ′)) , N j := P j(φ′) , (1)where the following following properties hold1. qi (N, φ) is defined on Rn × V−∞2. S : Rn × V−k 7→ S(N, φ) ∈ V l is smoothing.3. in terms of the variables φ′ the symplectic form is given byΩ(Φ′1; Φ′2) = 〈EΦ′1; Φ′2〉 . (2)The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryRecovering smoothnessThe differential of eqj (φ)JAjφ involveseqj (φ)JAjJAjφdqjSmoothness: If H is invariant: H(eqj (φ)JAju) = H(u) then:H(F(φ)) = H(φ+ smoothing terms)ExplicitelyH ◦ F = HL + HP + small correctionsHL(φ) = H(ηp0 + φ)− H(ηp0 )Restriction of the linearization at the soliton to the symplecticorthogonal!The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryOutline1 The Problem2 A model problem3 Recall of BC094 Connection with NLS5 Marsden-Weinstein reduction6 Darboux theorem7 Perturbation theoryThe Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theorySpectrumWrite HL(φ) =12 〈φ; Lφ〉One can prove that L is selfadjoint andSpectrum σ(L) ={ω21 , ..., ω2N} ∪ [Ω2,+∞)Coordinates There exist coordinates such thatHL =∑l ωl |ξl |2 +〈f¯ ;Bf〉One can start with perturbation theory.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryNormal form.Definition: Normal form A function Z (M, φ) is in normal form up toorder N, if the following holds{ω · (µ− ν) 6= 0 & |µ|+ |ν| ≤ N + 2} =⇒ ∂|µ|+|ν|Z∂ξµ∂ν ξ¯(M, 0) = 0{ω · (µ− ν) < Ω & |µ|+ |ν| ≤ N + 1} =⇒ df ∂|µ|+|ν|Z∂ξµ∂ν ξ¯(M, 0) = 0{−ω · (µ− ν) > Ω & |µ|+ |ν| ≤ N + 1} =⇒ df¯∂|µ|+|ν|Z∂ξµ∂ν ξ¯(M, 0) = 0TheoremFor any N ≥ 0 there exists a canonical transformation TN of the form (1)such that Hr ◦ TN is in normal form up to order N.The Problem A model problem Recall of BC09 Connection with NLS Marsden-Weinstein reduction Darboux theorem Perturbation theoryTHE ENDTHANK YOU

Asymptotic stability of solitons

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Asymptotic stability of solitons

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