Multisolitons for NLS

Transcription

1 Multisolitons for NLS Stefan LE COZ Beijing

2 Plan 1 Introduction 2 Existence of multi-solitons 3 (In)stability

3 NLS (NLS) { iut + u + g( u 2 )u = 0 u t=0 = u 0 u : R t R d x C (A0) (regular) g C 1 ((0, + ), R) (A1) (superlinear) g(0) = 0, lim s 0 sg (s) = 0 (A2) (H 1 -subcritical) p (1, 2 1) s.t. s 2 g (s 2 ) s p 1 (s > 1) (A3) (focusing) s 0 s.t. F (s 0 ) > s2 0 ; F (z) := 2 z 0 g(s 2 )sds.

4 Cauchy Problem Proposition (NLS) iu t + u + g( u 2 )u = 0 (NLS) is locally well-posed in H 1 under (A0)-(A3), i.e. there exists a unique maximal solution u C(( T, T ), H 1 ). Conserved Quantities E(u) = 1 2 u 2 L R 2 F (u)dx (Energy) d M(u) = 1 2 u 2 L 2 (Mass) P(u) = 1 2 Im R d ūudx (Momentum) Blow Up Alternative : If T < +, then lim t T u(t) H 1 = +

5 Large Time Behavior - Heuristic u solution of (NLS) iu t + u + g( u 2 )u = 0 3 possible behaviors at large time Scattering u(t) e i t u + as t + Focusing/Blow Up T < + Soliton fixed profile in a moving frame Solitons Resolution Conjecture Modest Goal Study solutions of NLS composed of several solitons

6 Solitons Definition of a soliton Take a frequency ω > 0, a speed v R d, an initial phase γ R, an initial position x 0 R d, and a bound state solution Φ H 1 to (SNLS) Φ + ωφ g( Φ 2 )Φ = 0, Φ H 1. A soliton is a solution of (NLS) traveling on the line x = x 0 + vt and given by R(t, x) := Φ(x vt x 0 )e i( 1 2 v x 1 4 v 2 t+ωt+γ).

7 Bound states (SNLS) Φ + ωφ g( Φ 2 )Φ = 0, Φ H 1 Proposition (Ground State) There exists a solution a Q of (SNLS), which minimizes the action S, i.e. S(Q) = min{s(w) w sol of (SNLS)} where S(w) := E(w) + ωm(w). (Excited States) If d 2, there exists infinitely many other solutions to (SNLS) (Exponential Decay) Any solution Φ to (SNLS) verifies Φ(x) e i ω 2 x. a unique if g( u 2 )u = u p 1 u

8 Dynamical properties of the solitons In the case g( u 2 )u = u p 1 u. with ground states 1 < p < d : stability (Cazenave-Lions, 1982) d p: instability (Berestycki-Cazenave, 1981; Weinstein, 1983) with excited states 1 < p < d : partial results of instability (Grillakis 1990, Mizumachi 2007, Chang, Gustafson, Nakanishi et Tsai 2007) d p: instability for real and radial excited states (Grillakis 1988, Jones 1988) or for some vortices (Mizumachi 2005).

9 Set ans sum of solitons Definition of a set of solitons A set of solitons is the data of (N, ω j, v j, x j, γ j, Φ j ), where N N, N 1, and for j = 1,..., N, we have ω j > 0, x j, v j R d with v j v k if j k, γ j R, and Φ j is a sol of (SNLS) (with ω j instead of ω). Given a set of solitons, we define the sum of solitons R(t, x) = N R j (t, x), j=1 where R j (t, x) := Φ j (x v j t x j )e i( 1 2 v j x 1 4 v j 2 t+ω j t+γ j ). NB : R is not a solution to (NLS)

10 Multi-solitons Definition of a multi-soliton A multisoliton is a solution u of (NLS) for which there exist T 0 and R defined as above such that u exists on [T 0, + ) and lim u(t) R(t) t + H 1 = 0. Natural questions about multisolitons Existence Uniqueness Stability

11 1972 Theorem (Zakharov and Shabat 1972) If d = 1, g( u 2 )u = u 2 u (completely integrable case), then for any set of solitons (N, ω j, v j, x j, γ j, Φ j ), there exists a solution u of (NLS) such that u exists on R there is an explicit construction for u u(t) R(t) 0 as t u(t) R(t) 0 as t +, where R R with ( x j, γ j ).

12 1990 Merle 1990 Let d 1 and take a set of solitons (N, ω j, v j, x j, γ j, Φ j ). Assume g( u 2 )u = u 4/d u (L 2 -critical case) and a large frequencies assumption. Then there exists T 0 > 0 and a multisoliton solution u of (NLS) s.t. lim u(t) R(t) t + H 1 = 0. Remark It is the dual version of the result of existence of a multiple blow-up points solutions (by conformal invariance).

13 2006 Theorem (Martel-Merle 2006) Let d 1 and take a set of solitons (N, ω j, v j, x j, γ j, Φ j ). Assume g( u 2 )u verifies a stability assumption (L 2 -subcritical) and (Φ j ) are ground states. Then there exist T 0 R and a solution u of (NLS) on [T 0, + ) s.t. u(t) R(t) H 1 e α ω v t on [T 0, + ) where α > 0 and v := min{ v j v k, j, k = 1,..., N, j k} (minimal relative speed) ω := min{ω j, j = 1,..., N} (minimal frequency)

14 2011 Theorem (Côte-Martel-Merle 2011) Let d 1 and take a set of solitons (N, ω j, v j, x j, γ j, Φ j ). Assume g( u 2 )u = u p 1 u, p > 1 + 4/d (L 2 supercritical) and (Φ j ) are ground states. Then there exist T 0 R and a solution u of (NLS) on [T 0, + ) s.t. u(t) R(t) H 1 e α ω v t on [T 0, + ) where α > 0 and v := min{ v j v k, j, k = 1,..., N, j k} (minimal relative speed) ω := min{ω j, j = 1,..., N} (minimal frequency)

15 2011 Theorem (Côte-L.C. 2011) Let d 1 and take a set of solitons (N, ω j, v j, x j, γ j, Φ j ). Assume g is generic (satisfies (A0)-(A3)) and (Φ j ) are ground states or excited states. There exists v such that if v > v (high relative speeds) then there exist T 0 R and a solution u of (NLS) on [T 0, + ) s.t. u(t) R(t) H 1 e α ω v t on [T 0, + ) where α > 0 and v := min{ v j v k, j, k = 1,..., N, j k} (minimal relative speed) ω := min{ω j, j = 1,..., N} (minimal frequency)

16 Summary for existence NLS 1972 Zakharov and Shabat, integrable case 1990 Merle, L 2 -critical case, high frequencies 2006 Martel-Merle, L 2 subcritical with ground states 2011 Côte-Martel-Merle, L 2 supercritical with ground states 2011 Côte-L.C., any nonlinearity, ground and excited states, high speeds Open problem Small speeds for excited states

17 Scheme of proof : Backward resolution of (NLS) Take a set of solitons (N, ω j, v j, x j, γ j, Φ j ). Take (T n ) + and (u n ) solutions to (NLS) with final data u n (T n ) = R(T n ). Goal Approximate Multisolitons. Show that for each n, u n exists on [T 0, T n ] with T 0 independent of n Convergence. Show that (u n ) converges to a multi-soliton Tools Uniform Estimates Compactness Argument

18 Tools Proposition (Uniform Estimates) There exists v > 0 s.t. if v > v there exists T 0 independent of n s.t. for n large u n exist on [T 0, T n ] and for all t [T 0, T n ] Proposition (Compactness) There exists u 0 H 1 s.t. u n (t) R(t) H 1 e α ω v t. lim u n(t 0 ) u 0 n + L 2 = 0 Rk: u 0 will be the initial data of the multisoliton.

19 Proof of the existence of a multisoliton Assume the uniform estimates and the compactness result. Take the solution u of (NLS) with at T 0 initial data u(t 0 ) = u 0. Then for t > T 0 L 2 un(t) u(t). H 1 Therefore u(t) R(t) H 1 lim inf u n(t) R(t) n + H 1 e α ω v t Hence u is a multi-soliton.

20 Proof of Uniform Estimates - Scheme A bootstrap argument Energy control of one soliton up to L 2 directions Localization procedure Energy control of N-solitons up to L 2 directions Control of the L 2 directions

21 The Bootstrap argument Proposition (Bootstrap) There exists v > 0 s.t. if v > v there exists T 0 independent of n s.t. for n large u n exist on [T 0, T n ] and verifies: for all t [T 0, T n ], if for all t [t, T n ] we have (Bootstrap-1) u n (t) R(t) H 1 e α ω v t, then for all t [t, T n ] we have (Bootstrap-1/2) u n (t) R(t) H e α ω v t.

22 Energy control of one soliton R 0 (t, x) := Φ 0 (x v 0 t x 0 )e i( 1 2 v 0 x 1 4 v 0 2 t+ω 0 t+γ 0 ). R 0 is a critical point of S 0 = E + (ω 0 + v 0 2 ) M + v 0 P 4 Define H 0 (t, w) := S 0 (R 0)w, w. Proposition (Coercivity) There exist ν 0 N \ {0} and X 1 0,..., X ν 0 0 L 2 s.t. for all w H 1 w 2 H 1 H ν 0 0(t, w) + (w, X0 k (t)) 2 2. k=1 where X k 0 (t)(x) = X k 0 (x v 0t x 0 )e i( 1 2 v 0 x 1 4 v 0 2 t+ω 0 t+γ 0 ).

23 Localization of the conservation laws - 1 For each j = 1,..., N, we define a cut-off function φ j :

24 Localization of the conservation laws - 2 and we set E j (t, w) = 1 w 2 φ j (t, x)dx F (w)φ j (t, x)dx 2 R d R d M j (t, w) = 1 w 2 φ j (t, x)dx 2 R d P j (t, w) = 1 2 R I w wφ j (t, x)dx d S j (t, w) =E j (t, w) + (ω j + v j 2 ) M j (t, w) v j P(t, w) 4 H j (t, w) = S j (R j (t))wφ j (t), w

25 Energy control for N solitons Proposition For any t large enough w n 2 H 1 H(t, w n ) + where H(t, w) = N j=1 H j(t, w). ν N j (w n, Xj l (t)) 2 2, j=1 l=1

26 Control of the linearized action Take w n = u n R and recall (Bootstrap-1) w n (t) H 1 e α ω v t on [t, T n ]. Control of the linearized action H(t, w n ) 1 t e 2αv t + o( w n 2 H 1 ) on [t, T n ]. Almost conservation of the localized quantities For T 0 large enough, we have on [t, T n ] and for any j M j (t, w n ) M j (T n, w n ) + P j (t, w n ) P j (T n, w n ) 1 t e 2α ω v t. Key ingredient : exponential decay of the solitons.

27 Control of the L 2 -directions Recall that w n = u n R and (Bootstrap-1): w n (t) H 1 e α ω v t on [t, T n ]. Then i t w n + Lw n + N (w n ) = 0, 1 d 2 dt w n 2 L 2 = (ilw n, w n ) 2 + (in (w n ), w n ) 2. It is easy to see that (ilw n, w n ) 2 C L 2 w n 2 L 2 ; (in (w n), w n ) 2 = o( w n 2 H ), 1 Then d dt w n 2 L 2t Ce 2αv t, donc w n 2 L 2 C 2αv e 2αv t.

28 Summary of the proof Backward resolution of (NLS) Uniform estimates Energy control Localization procedure Deal with L 2 -directions Compactness argument for the initial data

29 The notion of stability If a solution of (NLS) starts close to a sum of solitons, then it Orbital Stability: remains close for all time to the sum of solitons, possibly modified by translations or phase shifts. Asymptotic Stability: converges as t + to the sum of solitons, possibly modified by translations or phase shifts. (Forward) Instability: leaves in finite time the neighborhood of the sum of solitons, possibly modified by translations or phase shifts.

30 2004 Perelman 1997,2004/Rodnianski, Schlag, and Soffer 2003 If Flatness of the nonlinearity at 0, Spectral Assumptions (linear stability) High relative speeds. Then asymptotic stability of well-ordered multi-solitons in strong norms.

31 2006 Martel-Merle-Tsai 2006 If Flatness of the nonlinearity at 0, Orbital Stability of each solitons, High relative speeds. Then orbital stability of well-ordered multi-solitons in the energy space H 1.

32 A new construction of a multisoliton... Theoreme (Côte-L.C.) Take a R and a sum of solitons R = N j=1 R j. Assume g C and one soliton (e.g. R 1 ) is linearly unstable, i.e. there exists an eigenvalue λ C with R(λ) > 0 of the linearization L of (NLS) around R 1. Then there exists v > 0 s.t. if v > v (large relative speeds) there exist T 0 and u solution of (NLS) on [T 0, + ) s.t. for all t T 0 where Y is of the form u(t) R(t) ay (t) H 1 Ce 2R(λ)t Y (t) = e R(λ)t (cos(i(λ)t)y 1 + sin(i(λ)t)y 2 ).

33 ... which turns out to be unstable Corollary (Orbital instability of the multisoliton) Same hypotheses. There exists ε > 0, such that for all n N \ {0} and for all T R the following holds. There exists I n, J n R, T I n < J n and a solution w n C([I n, J n ], H 1 (R d )) to (NLS) such that inf y j R d,ϑ j R, j=1,...,n lim w n(i n ) R(I n ) H 1 = 0, n + N w n(j n ) i=1 Φ j (x y j )e i( 1 2 v j x+ϑ j ) L 2 ε.

34 Proof of the Theorem By fixed point around a good approximation of u. Proposition Take N 0 N and a R. Then there exists W N 0 C ([0, + ), H ) s.t. U = R + W N 0 is a solution to (NLS) up to an order O(e ρ(n0+1)t ) when t +, i.e. U t + U + g( U 2 )U = Err = O(e ρ(n0+1)t ) Define the map w Ψ(w) = i + t e i (t τ) (f ((U+w)(τ)) f (U(τ)) Err(τ))dτ. Fixed point argument in { } XT σ 0,N 0 (B) := w C((T 0, + ), H σ ) sup e (N0+1)ρt w(t) H σ < B. t T 0

35 Construction of the profile - preliminaries Inspired by works of Duyckaerts, Merle, Roudenko. Look at the linearization of (NLS) around e iωt Φ(x). If u is a solution of (NLS) and u = e iωt (Φ(x) + w), then v is a solution of w t + L C w = M C (w), where L C w = i w + iωw idf (Φ).w, M C (w) = if (Φ + w) if (Φ) idf (Φ).w,

36 Construction of the profil Proposition Take N 0 N and a R. Then there exists W N 0 C ([0, + ), H ) s.t. when t +, t W N 0 + L R 2W N 0 = M R 2(W N 0 ) + O(e ρ(n 0+1)t ), Separate L C into real and imaginary parts: ( ) ( ) ( ) wr J ω + I L R 2 = 1 wr. + ω I 2 J w I with I and J real valued potentials w I

37 Construction of the profile - order 1 Complexify L R 2 into L C 2 There exists an eigenvalue λ = ρ + iθ C of L C 2 with maximal real part ρ > 0. ( ) Z + Set Z = an eigenvector and denote Y 1 = R(Z), Y 2 = I(Z). Z Denote Y (t) = e ρt (cos(θt)y 1 + sin(θt)y 2 ). Then t Y + L R 2Y = 0.

38 Construction of the profile - order N 0 Look for W N 0 in the form N 0 k W (t, x) = e ρkt A j,k (x) cos(jθt) + B j,k (x) sin(jθt) Remark that k=1 N 0 M R 2(W ) = In addition κ=2 e κρt N 0 ( t W +L R 2W ) = k=1 κ j=0 j=0 (Ãj,κ (x) cos(jθt) + B j,κ (x) sin(jθt)) +HOT e ρkt k (L R 2A j,k + jθb j,k kρa j,k ) cos(jθt) j=0 + (L R 2B j,k jθa j,k kρb j,k ) sin(jθt) ).

39 Construction of the profile - order N 0 Therefore to find a satisfying W N 0 it is enought to solve for k 2 { LR 2A j,k + jθb j,k kρa j,k = Ã j,k, L R 2B j,k jθa j,k kρb j,k = B j,k, which is possible because λ is of maximal real part.

40 Open problems Existence for small speeds for excited states Better stability/instability results Uniqueness/Classification Other equations KdV : Martel, Merle, Muñoz, Combet Camassa-Holm : El Dika, Molinet Schrödinger systems : Ianni, L.C. Hartree : Krieger, Martel, Raphaël GP : Béthuel, Gravejat, Smets etc.