Hylomorphic solitons and their dynamics

Transcription

1 Hylomorphic solitons and their dynamics Vieri Benci Dipartimento di Matematica Applicata U. Dini Università di Pisa 18th May 2009 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

2 Types of waves Linear equations (with constant coefficients) produce waves which can be classified in two classes: Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

3 Types of waves Linear equations (with constant coefficients) produce waves which can be classified in two classes: non-dispersive waves dispersive waves. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

4 Types of waves Linear equations (with constant coefficients) produce waves which can be classified in two classes: non-dispersive waves dispersive waves. In the nonlinear case other very interesting phenomena may occur. solitary waves solitons vortices Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

5 Linear waves In the linear case, the dispersion phenomenon can be understood looking at the Fourier expansion of a solution: ψ (t, x) = a (k) e i(k x ωkt) dk ω k = ± k non-dispersive waves ω k = ± m 2 + k 2 dispersive waves Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

6 Linear waves In the linear case, the dispersion phenomenon can be understood looking at the Fourier expansion of a solution: ψ (t, x) = a (k) e i(k x ωkt) dk ω k = ± k non-dispersive waves ω k = ± m 2 + k 2 dispersive waves Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

7 Linear waves In the linear case, the dispersion phenomenon can be understood looking at the Fourier expansion of a solution: ψ (t, x) = a (k) e i(k x ωkt) dk ω k = ± k non-dispersive waves ω k = ± m 2 + k 2 dispersive waves Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

8 Non-dispersive waves: D Alembert equation D Alembert ψ = 0 (1) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

9 Dispersive waves: Klein-Gordon equation Klein-Gordon ψ + m 2 ψ = 0 (2) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

10 Nonlinear phenomena: solitary waves, solitons and vortices In this talk, we will use the following definitions: A solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

11 Nonlinear phenomena: solitary waves, solitons and vortices In this talk, we will use the following definitions: A solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behavior. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

12 Nonlinear phenomena: solitary waves, solitons and vortices In this talk, we will use the following definitions: A solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behavior. A vortex as a solitary wave with a non vanishing angular momentum. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

13 Standing solitary waves: NWE NWE ψ + ψ 1 + ψ = 0 (3) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

14 Travelling solitary waves Travelling waves ψ + ψ 1 + ψ = 0 (4) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

15 Vortices: NWE NWE ψ + ψ 1 + ψ = 0; ψ(t, x) = u(r)ei(θ ωt) (5) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

16 Vortices with l = 3 in NWE NWE ψ + ψ 1 + ψ = 0; ψ(t, x) = u(r)ei(3θ ωt) (6) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

17 Vortices with l = 8 in NWE NWE ψ + ψ 1 + ψ = 0; ψ(t, x) = u(r)ei(8θ ωt) (7) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

18 The beginning of the history of solitary waves and solitons is considered the year 1834 when Russell observed a solitary wave in a low water channel. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

19 Soliton types Today, we know (at least) three mechanism which might produce solitons: Complete integrability; e.g. Kortewg-de Vries equation u t + u xxx + 6uu x = 0 Topological constrains: e.g. Sine-Gordon equation u tt u xx + sin u = 0 Ratio energy/charge: e.g. complex valued nonlinear Klein-Gordon equation (they exists also for N > 1) ψ tt ψ + W (ψ) = 0; ψ C Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

20 Soliton types Today, we know (at least) three mechanism which might produce solitons: Complete integrability; e.g. Kortewg-de Vries equation u t + u xxx + 6uu x = 0 Topological constrains: e.g. Sine-Gordon equation u tt u xx + sin u = 0 Ratio energy/charge: e.g. complex valued nonlinear Klein-Gordon equation (they exists also for N > 1) ψ tt ψ + W (ψ) = 0; ψ C Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

21 Soliton types Today, we know (at least) three mechanism which might produce solitons: Complete integrability; e.g. Kortewg-de Vries equation u t + u xxx + 6uu x = 0 Topological constrains: e.g. Sine-Gordon equation u tt u xx + sin u = 0 Ratio energy/charge: e.g. complex valued nonlinear Klein-Gordon equation (they exists also for N > 1) ψ tt ψ + W (ψ) = 0; ψ C Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

22 Soliton types Today, we know (at least) three mechanism which might produce solitons: Complete integrability; e.g. Kortewg-de Vries equation u t + u xxx + 6uu x = 0 Topological constrains: e.g. Sine-Gordon equation u tt u xx + sin u = 0 Ratio energy/charge: e.g. complex valued nonlinear Klein-Gordon equation (they exists also for N > 1) ψ tt ψ + W (ψ) = 0; ψ C Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

23 Soliton types The complete integrability solitons occur only in dimension one (at least for what I know). Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

24 Soliton types The complete integrability solitons occur only in dimension one (at least for what I know). Topological solitons may occur also in dimension bigger than one but not in semilinear equations. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

25 Soliton types The complete integrability solitons occur only in dimension one (at least for what I know). Topological solitons may occur also in dimension bigger than one but not in semilinear equations. The third type of solitons may appear in any dimension. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

26 Complete integrability solitons This kind of solitons occur in infinite dimensional completely integrable systems. Their study is connected with the inverse scattering techniques and it is very involved. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

27 Topological solitons This kind of solitons due their stability to topological constrains. The simplest equation which produces topological solitons is the Sine-Gordon equation (which is also completely integrable) u tt u xx + sin u = 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

28 The Hylomorphic solitons The third kind of solitons due their existence to the low ratio between the energy and an other integral of motion which we call hylenic charge Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

29 The Hylomorphic solitons The third kind of solitons due their existence to the low ratio between the energy and an other integral of motion which we call hylenic charge I will refer to this type of solitons as hylomorphic solitons Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

30 The Hylomorphic solitons The third kind of solitons due their existence to the low ratio between the energy and an other integral of motion which we call hylenic charge I will refer to this type of solitons as hylomorphic solitons In the rest of the talk, we will be concerned with this type of solitons Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

31 History of hylomorphic solitons We introduced the name hylomorphic soliton to denote various kinds of solitons already studied in the mathematical and physical literature in order to emphasize their commune features (namely the ratio energy/charge). Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

32 History of hylomorphic solitons We introduced the name hylomorphic soliton to denote various kinds of solitons already studied in the mathematical and physical literature in order to emphasize their commune features (namely the ratio energy/charge). The mathematical and physical communities have studied hylomorphic solitons quite independently. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

33 History of hylomorphic solitons We introduced the name hylomorphic soliton to denote various kinds of solitons already studied in the mathematical and physical literature in order to emphasize their commune features (namely the ratio energy/charge). The mathematical and physical communities have studied hylomorphic solitons quite independently. In particular, the physicists considered mainly the nonlinear Klein-Gordon equations and the relative solitons were called Q-balls. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

34 History of Q-balls Configurations of a charged scalar field that are classically stable were constructed by G. Rosen who first realized the possibility of stable solitary waves in the NWE. G. ROSEN,Particlelike Solutions to Nonlinear Complex Scalar Field Theories with Positive-Definite Energy Densities, J. Math. Phys. 9:996 (1968) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

35 History of Q-balls Configurations of a charged scalar field that are classically stable were constructed by G. Rosen who first realized the possibility of stable solitary waves in the NWE. G. ROSEN,Particlelike Solutions to Nonlinear Complex Scalar Field Theories with Positive-Definite Energy Densities, J. Math. Phys. 9:996 (1968) The name Q-ball came from Sidney Coleman S. COLEMAN, Q-BALLS, Nucl. Phys. B262:263 (1985); erratum: B269:744 (1986) For the moment, the main physical applications of Q-balls are in cosmology. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

36 History of hylomorphic solitons In the mathematical community this story began with a paper of Strauss relative to solitary waves: W.A. STRAUSS, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

37 History of hylomorphic solitons Other important contributions during the 80 s: BERESTYCKI H., LIONS P.L., Nonlinear scalar field equations, I - Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), T. CAZENAVE, P.L. LIONS, Orbital Stability of Standing Waves for Some Non linear Schrödinger Equations, Commun. Math. Phys. 85, (1982) J. SHATAH, Stable Standing waves of Nonlinear Klein-Gordon Equations, Comm. Math. Phys., 91, (1983), Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

38 Some recent results on the nonlinear wave equation BENCI V. FORTUNATO D., Solitary waves in the Nolinear Wave equation and in Gauge Theories, Journal of fixed point theory and Applications, 1, n.1 (2007) p.p J. BELLAZZINI, V. BENCI, C. BONANNO, A.M. MICHELETTI, Solitons for the Nonlinear Klein-Gordon-Equation, to appear J. BELLAZZINI, V. BENCI, C. BONANNO, E. SINIBALDI, Hylomorphic solitons, to appear Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

39 The hylomorphic solitons The main equations in which hylomorphic solitons may occur are the following: Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

40 The hylomorphic solitons The main equations in which hylomorphic solitons may occur are the following: Nonlinear Scroedinger equation: ih t ψ = h2 2 ψ W (ψ) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

41 The hylomorphic solitons The main equations in which hylomorphic solitons may occur are the following: Nonlinear wave equation: ψ + W (ψ) = 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

42 Gauge theories e.g. the Maxwell-Klein-Gordon equations: D 2 t ψ 3 D 2 x j ψ + W (ψ) = 0 (8) j=1 E = ρ (ψ, ϕ) ; E + H t = 0; (9) H E t = j (ψ, A) ; H = 0; (10) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

43 Gauge theories e.g. the Maxwell-Klein-Gordon equations: D 2 t ψ 3 D 2 x j ψ + W (ψ) = 0 (8) j=1 E = ρ (ψ, ϕ) ; E + H t = 0; (9) H E t = j (ψ, A) ; H = 0; (10) where D t = t + iqϕ, D xj = xj iqa j, ρ (ψ, ϕ) = q Im ( t ψψ ) + q 2 ϕ ψ 2 (11) j (ψ, A) = q Im ( ψψ ) q 2 A ψ 2 (12) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

44 These equations satisfy the two assumptions which are shared by every fundamental theory in Physics. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

45 These equations satisfy the two assumptions which are shared by every fundamental theory in Physics. These two assumptions are the following ones: A-1. They are variational. A-2. They are invariant for an action the Poincaré (or the Galileo) group. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

46 These equations satisfy the two assumptions which are shared by every fundamental theory in Physics. These two assumptions are the following ones: A-1. They are variational. A-2. They are invariant for an action the Poincaré (or the Galileo) group. Moreover they satisfy also the following assumption: A-3. They are invariant for the action of the group S 1 : T ϕ ψ e iϕ ψ. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

47 Conservation laws The invariance with respect to the action of a Lie group gives conservation Laws: Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

48 Conservation laws The invariance with respect to the action of a Lie group gives conservation Laws: Energy. Energy, by definition, is the quantity which is preserved by the time invariance of the Lagrangian; namely by the fact that it does not depend explicitly on t. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

49 Conservation laws The invariance with respect to the action of a Lie group gives conservation Laws: Energy. Energy, by definition, is the quantity which is preserved by the time invariance of the Lagrangian; namely by the fact that it does not depend explicitly on t. Hylenic Charge. The hylenic charge, by definition, is the quantity which is preserved by by the "gauge action" ψ ψe iθ ; namely by the fact that the Lagrangian does not depend explicitly on θ. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

50 The Nonlinear Schroedinger equation For example, let us consider the NSE: hi t ψ = h2 2 ψ W (ψ) The solution of this equations are critical points of the functional S(ψ) = [Re ( hi t ψψ ) ] h2 2 ψ 2 W(ψ) dx dt Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

51 Conservation laws for the NSE Energy. [ ] h 2 E(ψ) = 2 ψ 2 + W(ψ) dx Hylenic charge. H = ψ 2 dx Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

52 The Nonlinear Wave Equation As second example, we consider the NWE: ψ + W (ψ) = 0 The solution of this equations are critical points of the functional S(ψ) = [ 1 2 tψ 2 1 ] 2 ψ 2 W(ψ) dx dt (NWE) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

53 Conservation laws for the NWE Energy. E = [ 1 2 tψ ] 2 ψ 2 + W(ψ) dx Hylenic charge. H = Im t ψψ dx Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

54 Definition of hylomorphic soliton Since the notion of soliton is based on the notion of stability, we need to consider the dynamical system (X, U) relative to our equation. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

55 Definition of hylomorphic soliton Since the notion of soliton is based on the notion of stability, we need to consider the dynamical system (X, U) relative to our equation. If Ψ 0 (x) X, then the dynamics is given by (U t Ψ 0 ) (x) = Ψ(t, x) where Ψ(t, x) is the solution of the Cauchy problem with initial data Ψ 0 (x). Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

56 Definition of hylomorphic soliton Since the notion of soliton is based on the notion of stability, we need to consider the dynamical system (X, U) relative to our equation. If Ψ 0 (x) X, then the dynamics is given by (U t Ψ 0 ) (x) = Ψ(t, x) where Ψ(t, x) is the solution of the Cauchy problem with initial data Ψ 0 (x). For NSE we have that X = H 1 (R N ) For NWE we have that X = H 1 (R N ) L 2 (R N ) Ψ(x) = (ψ(x), ψ t (x)) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

57 Definition of hylomorphic soliton A standing wave Ψ 0 X is a hylomorphic soliton if where E(Ψ 0 ) = min E(Ψ) Ψ M σ M σ = {Ψ X : H(Ψ) = σ} The set Γ of minimizer of E(Ψ) on M σ is a finite dimensional manifold (or, if you want to be more general, a locally compact set) The set Γ (which is an invariant set for U t ) is Liapunov stable (namely Ψ(t, x) is orbitally stable). Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

58 Definition of hylomorphic soliton A standing wave Ψ 0 X is a hylomorphic soliton if where E(Ψ 0 ) = min E(Ψ) Ψ M σ M σ = {Ψ X : H(Ψ) = σ} The set Γ of minimizer of E(Ψ) on M σ is a finite dimensional manifold (or, if you want to be more general, a locally compact set) The set Γ (which is an invariant set for U t ) is Liapunov stable (namely Ψ(t, x) is orbitally stable). Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

59 Definition of hylomorphic soliton A standing wave Ψ 0 X is a hylomorphic soliton if where E(Ψ 0 ) = min E(Ψ) Ψ M σ M σ = {Ψ X : H(Ψ) = σ} The set Γ of minimizer of E(Ψ) on M σ is a finite dimensional manifold (or, if you want to be more general, a locally compact set) The set Γ (which is an invariant set for U t ) is Liapunov stable (namely Ψ(t, x) is orbitally stable). Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

60 Definition of hylomorphic soliton A standing wave Ψ 0 X is a hylomorphic soliton if where E(Ψ 0 ) = min E(Ψ) Ψ M σ M σ = {Ψ X : H(Ψ) = σ} The set Γ of minimizer of E(Ψ) on M σ is a finite dimensional manifold (or, if you want to be more general, a locally compact set) The set Γ (which is an invariant set for U t ) is Liapunov stable (namely Ψ(t, x) is orbitally stable). Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

61 Existence of standing waves for NSE and NWE If W is written as follows W (u) = 1 2 au2 + N(u); a > 0; N(0) = N (0) = N (0) = 0 then sufficient (and essentially necessary) conditions for the existence of standing waves (Beresticky-Lions) are: Growth condition N (u) c u p 1, 2 p < 2 = 2N N 2 Hylomorphy condition u 0 R + : N(u 0 ) < 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

62 Existence of solitons for NSE Sufficient (and essentially necessary) conditions for the existence of solitons are (Cazenave-Lions; Bellazzini, Ghimenti, B., Micheletti) are: (Cauchy problem) the Cauchy problem is well posed in X namely ψ (t, x) C ( [0, T], H 1 (R N ) ) (Growth condition) N (u) c u p 1, 2 p < N Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

63 Existence of solitons for NSE Sufficient (and essentially necessary) conditions for the existence of solitons are (Cazenave-Lions; Bellazzini, Ghimenti, B., Micheletti) are: (Cauchy problem) the Cauchy problem is well posed in X namely ψ (t, x) C ( [0, T], H 1 (R N ) ) (Growth condition) N (u) c u p 1, 2 p < N Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

64 Existence of solitons for NWE Sufficient (and essentially necessary) conditions for the existence of solitons are (Bellazzini, Bonanno, B., Micheletti) are: (Cauchy problem) the Cauchy problem is well posed in X namely ψ (t, x) C ( [0, T], H 1 (R N ) ) C 1 ( [0, T], L 2 (R N ) ) (Positivity) u, W(u) 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

65 Existence of solitons for NWE Sufficient (and essentially necessary) conditions for the existence of solitons are (Bellazzini, Bonanno, B., Micheletti) are: (Cauchy problem) the Cauchy problem is well posed in X namely ψ (t, x) C ( [0, T], H 1 (R N ) ) C 1 ( [0, T], L 2 (R N ) ) (Positivity) u, W(u) 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

66 Idea of the proof By the hylomorphy condition, in a bump solution, the ratio is less than some constant m energy/carge Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

67 Idea of the proof By the hylomorphy condition, in a bump solution, the ratio energy/carge is less than some constant m while for "small" waves this ratio is greater than m. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

68 Dynamics of solitons (NSE) The basic idea is that a sufficiently small soliton behaves as a particle. In particular a soliton relative to NSE ih ψ t = h2 2 ψ + V(x)ψ W h(ψ) behaves as a classical particle as h 0; Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

69 Dynamics of solitons (NSE) The basic idea is that a sufficiently small soliton behaves as a particle. In particular a soliton relative to NSE ih ψ t = h2 2 ψ + V(x)ψ W h(ψ) behaves as a classical particle as h 0; namely its position q(t) must be an approximate solution of the equation q + V(q h ) = 0. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

70 The nonlinear term W h We choose W h (u) = 1 h α+γ W (hγ u). Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

71 The nonlinear term W h We choose W h (u) = 1 h α+γ W (hγ u). The relative elliptic equation is h2 2 u + 1 2h α W (h γ u) = ωu Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

72 The nonlinear term W h We choose W h (u) = 1 h α+γ W (hγ u). The relative elliptic equation is and the ground state has the form u h (x) = 1 ( x ) h γ U h β h2 2 u + 1 2h α W (h γ u) = ωu with β = 1 + α γ 2 where U : R N, N 2, is a positive, radially symmetric solution of the static Nonlinear Schroedinger equation equation with U L 2 = 1 U + W (U) = 2ωU (13) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

73 The Cauchy problem with V(x)=0 Now, let us consider the Cauchy problem relative to the NSE: ih ψ t = h2 2 ψ + 1 2h α W (h γ ψ) (14) ψ (0, x) = 1 ( ) x h γ U q0 h β e i h v x ; β = 1 + α γ 2 (15) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

74 The Cauchy problem with V(x)=0 Now, let us consider the Cauchy problem relative to the NSE: ih ψ t = h2 2 ψ + 1 2h α W (h γ ψ) (14) ψ (0, x) = 1 ( ) x h γ U q0 h β e i h v x ; β = 1 + α γ 2 (15) Direct computations show that a solution of (14),(15) is given by with ψ (t, x) = 1 ( ) x h γ U q0 vt h β e i h (v x Et) (16) E = 1 2 v2 + ω h α γ Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

75 The barycenter We can interprete this result saying that the barycenter q(t) of the solution of (14,15) defined by q(t) = satisfies the Cauchy problem x ψ(t, x) 2 dx RN. (17) ψ(t, x) 2 dx R N q = 0 q(0) = q 0 q(0) = v Thus the soliton is like a material point which follows Newton s laws. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

76 The Cauchy problem with V(x) 0 Let us investigate what happens if the problem is perturbed namely let us investigate the Cauchy problem ih ψ t = h2 1 ψ + V(x)ψ + 2 2h α W (h γ ψ) (P h ) with initial conditions where ψ (0, x) = [ 1 h γ U ( x x0 h β ) ] + w 0 (x) e i h v x (I h ) w 0 H 1 = 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

77 General assumptions (i) the problem (P h ),(I h ) has a unique solution ψ (t, x) C 0 ( R, H 1 (R N ) ) C 1 ( R, L 2 (R N ) ) (sufficient conditions have been found by Kato, Cazenave, Ginibre-Velo, etc.). (ii) the unperturbed equation admits hylomorphic solitons, namely there exists a minimizer U of the functional 1 J(u) = 2 u 2 + W(u)dx on the manifold S σ = M σ = { u H 1 : u L 2 = σ }. (iii) V(x) C 2 (R N ); V(x) 0. (iv) V(x) V(x) 1 δ for x > R 1 > 1, and some δ > 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

78 General assumptions (i) the problem (P h ),(I h ) has a unique solution ψ (t, x) C 0 ( R, H 1 (R N ) ) C 1 ( R, L 2 (R N ) ) (sufficient conditions have been found by Kato, Cazenave, Ginibre-Velo, etc.). (ii) the unperturbed equation admits hylomorphic solitons, namely there exists a minimizer U of the functional 1 J(u) = 2 u 2 + W(u)dx on the manifold S σ = M σ = { u H 1 : u L 2 = σ }. (iii) V(x) C 2 (R N ); V(x) 0. (iv) V(x) V(x) 1 δ for x > R 1 > 1, and some δ > 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

79 General assumptions (i) the problem (P h ),(I h ) has a unique solution ψ (t, x) C 0 ( R, H 1 (R N ) ) C 1 ( R, L 2 (R N ) ) (sufficient conditions have been found by Kato, Cazenave, Ginibre-Velo, etc.). (ii) the unperturbed equation admits hylomorphic solitons, namely there exists a minimizer U of the functional 1 J(u) = 2 u 2 + W(u)dx on the manifold S σ = M σ = { u H 1 : u L 2 = σ }. (iii) V(x) C 2 (R N ); V(x) 0. (iv) V(x) V(x) 1 δ for x > R 1 > 1, and some δ > 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

80 General assumptions (i) the problem (P h ),(I h ) has a unique solution ψ (t, x) C 0 ( R, H 1 (R N ) ) C 1 ( R, L 2 (R N ) ) (sufficient conditions have been found by Kato, Cazenave, Ginibre-Velo, etc.). (ii) the unperturbed equation admits hylomorphic solitons, namely there exists a minimizer U of the functional 1 J(u) = 2 u 2 + W(u)dx on the manifold S σ = M σ = { u H 1 : u L 2 = σ }. (iii) V(x) C 2 (R N ); V(x) 0. (iv) V(x) V(x) 1 δ for x > R 1 > 1, and some δ > 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

81 The main theorem Theorem (B.,Ghimenti,Micheletti) Assume that (i), (ii), (iii) and (iv) hold. Moreover assume that V(x) x α for x > R 1 > 1, α > 1; α γ > 0 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

82 The main theorem Theorem (B.,Ghimenti,Micheletti) Assume that (i), (ii), (iii) and (iv) hold. Moreover assume that V(x) x α for x > R 1 > 1, α > 1; α γ > 0 Then the barycenter q h (t) of the solution of the problem (P h ),(I h ) is twice differentiable and satisfies the following Cauchy problem: where q h (t) + V(q h (t)) = H h (t) q h (0) = q 0 q h (0) = v sup H h (t) 0 as h 0 t R (18) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

83 Bibliography on this subject J. C. Bronski and R. L. Jerrard, Soliton dynamics in a potential, Math. Res. Lett. 7 (2000), no. 2-3, Jürg Fröhlich, Stephen Gustafson, B. Lars G. Jonsson, and Israel Michael Sigal, Solitary wave dynamics in an external potential, Comm. Math. Phys. 250 (2004), no. 3, Jürg Fröhlich, Stephen Gustafson, B. Lars G. Jonsson, and Israel Michael Sigal, Long time motion of NLS solitary waves in a confining potential, Ann. Henri Poincaré 7 (2006), no. 4, They all considered the case α = γ = 0 and have some results only for finite time. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

84 The assumption V(x) The assumption of a confining potential is necessary in order to identify the "position" of the soliton with the barycenter of the full system. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

85 The assumption V(x) The assumption of a confining potential is necessary in order to identify the "position" of the soliton with the barycenter of the full system. If this assumption is violated, it is necessary to give a different definition of q(t), the "position" of the soliton. We get a similar result, but the technical part is more involved and we are still working on it. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

86 Meaning of α, γ, β = 1 + α γ 2 If ψ h is a soliton, then ψ h L 1 h γ Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

87 Meaning of α, γ, β = 1 + α γ 2 If ψ h is a soliton, then ψ h L 1 h γ "radius" of ψ h h β Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

88 Meaning of α, γ, β = 1 + α γ 2 If ψ h is a soliton, then ψ h L 1 h γ "radius" of ψ h h β ψ h p L p hnβ pγ Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

89 So, if we set we have that p = Nβ γ, (19) ψ h L p = const (20) and ψ h (t, x) p δ q(t) as h 0 (21) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

90 So, if we set we have that p = Nβ γ, (19) ψ h L p = const (20) and ψ h (t, x) p δ q(t) as h 0 (21) But it is also possible to choose α or γ equal to 0. In particular, if γ = 0, we have that ψ h L = const (22) Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

91 α γ > 0 E (ψ) = ( ) h 2 2 ψ 2 + V(x) ψ 2 dx + = E L (ψ) + E B (ψ) 1 h α+γ W(hγ ψ)dx If ψ h is a soliton, then, its linear energy is given by E L (ψ h ) ψ h 2 L 2 hnβ 2γ while its binding energy E B (ψ h ) h Nβ α γ. Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

92 Then E L (ψ h ) E B (ψ h ) hα γ This means that, as h 0, the binding energy is much stronger than the external energies: Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

93 Then E L (ψ h ) E B (ψ h ) hα γ This means that, as h 0, the binding energy is much stronger than the external energies: the soliton cannot be broken Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50

94 The end Collision of solitons Happy birthday Paul!!! Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May / 50