Asymptotic stability for solitons of the Gross-Pitaevskii and Landau-Lifshitz equations Philippe Gravejat Cergy-Pontoise University Joint work with F. Béthuel (Paris 6), A. de Laire (Lille) and D. Smets (Paris 6), and by Y. Bahri (Nice).
Introduction 1. The Gross-Pitaevskii equation The Gross-Pitaevskii equation writes as for a function Ψ : R N R C. i t Ψ + Ψ + Ψ(1 Ψ 2 ) = 0, The equation is Hamiltonian. Its Hamiltonian is the so-called Ginzburg-Landau energy, which is given by E GP (Ψ) := 1 2 R N Ψ 2 + 1 4 R N(1 Ψ 2 ) 2. In the sequel, we restrict the analysis to the Hamiltonian framework of finite energy solutions.
The equation is also dispersive. Its dispersion relation is equal to ω 2 GP = k 4 + 2 k 2. The norm of the group velocity is at least equal to the sound speed c s = 2. When the map Ψ does not vanish, it may be lifted as Ψ = 1 η exp iϕ. The functions η and v := ϕ are solutions of the hydrodynamical Gross-Pitaevskii equation t η = 2div ( (1 η)v ), t v = ( η v 2 η 2(1 η) ) η 2 4(1 η) 2, which is related to the compressible Euler equation.
2. The Landau-Lifshitz equation The Landau-Lifshitz equation ) t m + m ( m λm 3 e 3 = 0, (LL) describes the dynamics of the magnetization m = (m 1, m 2, m 3 ) : R N R S 2 in a ferromagnetic material. The vector e 3 = (0, 0, 1) and the parameter λ give account of the anisotropy of the material. In the sequel, we focus on the easy-plane anisotropy along the plane x 3 = 0 by setting λ = 1.
The Landau-Lifshitz equation is Hamiltonian. Its Hamiltonian is the Landau-Lifshitz energy, which is equal to E LL (m) := 1 ( ) m 2 + m 2 2 R N 3. The equation is also dispersive. The dispersion relation is given by the formula ω 2 LL = k 4 + k 2. This relation is almost identical to the dispersion relation of the Gross-Pitaevskii equation.
When the map ˇm := m 1 + im 2 does not vanish, it may be lifted as ˇm = 1 m 2 3 exp iϕ. The functions v := m 3 and w := ϕ solve the hydrodynamical Landau-Lifshitz equation t v = div ( (1 v 2 )w ), t w = ( v v w 2 v 1 v 2 v v 2 (1 v 2 ) 2 ). (HLL) which is similar to the hydrodynamical Gross-Pitaevskii equation.
3. Travelling-wave solutions Travelling waves are special solutions of the form m(x, t) = m c (x 1 c t,..., x N ). Their profile m c is solution to the nonlinear elliptic equation m c + m c 2 m c + [m c ] 2 3 m c [m c ] 3 e 3 + cm c 1 m c = 0. In dimension N = 1, travelling-wave solutions are called dark solitons. They are explicitly given by the following formulae.
Lemma. There exist non-constant solitons with speed c if and only if c < 1. Up to translations, rotations around the axis x 3, and the orthogonal symmetry with respect to the plane x 3 = 0, they are given by the formulae [m c ] 1 (x) = c ( cosh ( 1 c 2 x ), [m c ] 2 (x) = tanh 1 c 2 x ), and [m c ] 3 (x) = 1 c 2 cosh ( 1 c 2 x ).
When c 0, the soliton m c = ( 1 [m c ] 2 3 cos(ϕ c), 1 [m c ] 2 3 sin(ϕ c), [m c ] 3 ), can be identified in the hydrodynamical formulation with the pair where Q c := ( v c, w c ) = ( [mc ] 3, x ϕ c ), and v c (x) = w c (x) = (1 c 2 ) 1 2 cosh((1 c 2 ) 1 2x ), c v c(x) 1 v c (x) 2.
4. Integrability by the inverse scattering method In dimension N = 1, the Gross-Pitaevskii equation (Zakharov- Shabat [73]) and the Landau-Lifshitz equation (Sklyanin [79]) are integrable by means of the inverse scattering method. Given a solution Ψ of the Gross-Pitaevskii equation, the operators ( ) i(1 + 3) x Ψ L Ψ = Ψ i(1, 3) x and satisfy B Ψ = 3 2 xx + Ψ 2 1 i x Ψ 3+1 i x Ψ 3 2 xx + Ψ 2 1 3 1 t L Ψ(t) = i[l Ψ(t), B Ψ(t) ].,
The spectral characteristics of the operators L Ψ(t) are conserved along the flow. The solution Ψ(t) can be recovered using the spectral characteristics of the operator L Ψ(t). This provides an (at least formal) description of the long-time dynamics, which is governed by the propagation of a train of solitons plus a dispersive part (see Gérard-Zhifei Zhang [08] and Cuccagna-Jenkins [16]).
I. The Cauchy problem 1. In the original framework The energy space is defined as E GP (R) = { ψ : R C, s.t. x ψ L 2 (R) and 1 ψ 2 L 2 (R) }. Theorem (Zhidkov [01]). Let Ψ 0 E GP (R). There exists a unique global solution Ψ C 0 (R, E GP (R)) to (GP) with initial datum Ψ 0. The Ginzburg- Landau energy is conserved along the flow. (See Zhou-Guo [84], Sulem-Sulem-Bardos [86], Zhou-Guo-Tan [91], Chang-Shatah-Uhlenbeck [00], McGahagan [04] and Nahmod-Shatah-Vega-Zeng [06])
2. In the hydrodynamical framework The non-vanishing space is defined as N V(R) = { (η, v) H 1 (R) L 2 (R), s.t. max x R η(x) < 1}. Theorem (Mohamad [12]). Let (η 0, v 0 ) N V(R). There exists a maximal time T max > 0 and a unique solution (η, v) C 0 ([0, T max ), N V(R)) to (HGP) with initial datum (η 0, v 0 ). The maximal time T max is characterized by lim t T max max x R η(x, t) = 1 if T max < +. The Ginzburg-Landau energy and the momentum are conserved along the flow. (See de Laire-G. [15])
II. Orbital stability of (trains of) solitons 1. Minimizing nature of solitons For c 0, the soliton Q c is a minimizer of the variational problem E min (p c ) = inf { E LL (v), v : R C s.t. P LL (v) = p c }. The speed c is related to the momentum p c by the formula ( ) 1 c p c = arctan. c The minimizing energy is equal to E min (p c ) = 2 1 c 2.
2. Orbital stability of a well-prepared train of solitons Let a R N, c ( 1, 1) N and s {±1} N. A train of solitons is defined as a perturbation of a sum of solitons S c,a,s (x) := N j=1 s j Q cj (x a j ) = N j=1 s j ( vcj (x a j ), w cj (x a j ) ). The train is well-prepared when the speeds are ordered according to the positions, i.e. if when c 1 < < c N, a 1 < < a N.
Theorem (de Laire-G. [15]). Let c 0 ( 1, 1) N, s 0 {±1} N, and v 0 = (v 0, w 0 ) N V(R) such that c 0 1 < < 0 < < c0 N. There exist four positive numbers α, ν, A and L, such that, if v 0 H S c 0,a 0,s 0 1 L 2 = α0 < α, for positions a 0 R N such that min { a 0 k+1 a0 k, 1 k N 1} = L 0 > L,
then there exists a unique solution v C 0 (R +, N V(R)) of (HLL) with initial datum v 0, as well as N functions a k C 1 (R +, R), with a k (0) = a 0 k, such that a k (t) c0 k A ( α 0 + e ν L 0), and v(, t) S c 0,a(t),s 0 H 1 L 2 A ( α 0 + e ν L 0), for any non-negative number t. (See Zhiwu Lin [02], Béthuel-G.-Saut-Smets [08], Gérard-Zhifei Zhang [08], and Béthuel-G.-Smets [14]) The proof relies on arguments developed by Martel-Merle-Tsai [02, 06] in the spirit of the strategy by Weinstein [85] and Grillakis-Shatah-Strauss [87] for a single soliton.
III. Asymptotic stability of solitons 1. In the hydrodynamical framework Theorem (Béthuel-G.-Smets [15]). Let 0 < c < 2 and (η 0, v 0 ) N V(R). There exists a number α > 0 such that, if η 0 H v η c 1 + 0 L v c 2 < α, then, there exist a unique solution (η, v) C 0 (R, N V(R)) to (HGP) with initial datum (η 0, v 0 ), as well as a speed 0 < c < 2 and a position function a C 1 (R, R) such that ( η( + a(t), t), v( + a(t), t) ) Qc in H 1 (R) L 2 (R), and as t +. a (t) c, (See Bahri [16, 16])
2. In the original framework Theorem (Béthuel-G.-Smets [15], G.-Smets [15]). Let c ( 2, 2) and Ψ 0 E(R). There exists a number δ > 0 such that, if d c (Ψ 0, U c ) < δ, then, there exist a number c ( 2, 2), and two functions a C 1 (R, R) and θ C 1 (R, R) such that the solution Ψ to (GP) with initial datum Ψ 0 satisfies e iθ(t) Ψ( + a(t), t) U c in L loc (R), e iθ(t) x Ψ( + a(t), t) x U c in L 2 (R), and as t +. a (t) c, θ (t) 0,
3. Sketch of the proof of asymptotic stability The proof relies on arguments developed by Martel-Merle [00, 01, 05, 08] for the Korteweg-de Vries equation. It splits into three main steps : 1. constructing a limit profile, 2. establishing its smoothness and exponential decay, 3. proving that a smooth localized solution in the neighborhood of a soliton is a soliton.
Thank you very much!