Spatial decay of rotating waves in parabolic systems

Transcription

1 Spatial decay of rotating waves in parabolic systems Nonlinear Waves, CRC 701, Bielefeld, June 19, 2013 Denny Otten Department of Mathematics Bielefeld University Germany June 19, 2013 CRC 701 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

2 Outline 1 Introduction: Rotating pattern in R d 2 Main result: Exponential decay of v 3 Outline of proof: Exponential decay of v 4 Spectrum of rotating waves Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

3 Rotating Patterns in R d Consider a reaction diffusion system (1) u t (x,t) = A u(x,t)+f(u(x,t)), t > 0, x R d, d 2, u(x,0) = u 0 (x), t = 0, x R d. where u : R d [0, [ R N, A R N,N, f C 2 (R N,R N ). Assume a rotating wave solution u : R d [0, [ R N of (1) u (x,t) = v (e ts x) v : R d R N profile (pattern), 0 S R d,d skew-symmetric. Transformation (into a rotating frame): v(x,t) = u(e ts x,t) solves (2) v t (x,t) = A v(x,t)+ Sx, v(x,t) +f(v(x,t)), t > 0, x R d, d 2, v(x,0) = u 0 (x), t = 0, x R d. Sx, v(x) := d d S ij x j D i v(x) (drift term). i=1 j=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

4 Rotating Patterns in R d Consider a reaction diffusion system (1) u t (x,t) = A u(x,t)+f(u(x,t)), t > 0, x R d, d 2, u(x,0) = u 0 (x), t = 0, x R d. where u : R d [0, [ R N, A R N,N, f C 2 (R N,R N ). Assume a rotating wave solution u : R d [0, [ R N of (1) u (x,t) = v (e ts x) v : R d R N profile (pattern), 0 S R d,d skew-symmetric. Transformation (into a rotating frame): v(x,t) = u(e ts x,t) solves (2) v t (x,t) = A v(x,t)+ Sx, v(x,t) +f(v(x,t)), t > 0, x R d, d 2, v(x,0) = u 0 (x), t = 0, x R d. Sx, v(x) := d d S ij x j D i v(x) (drift term). i=1 j=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

5 Rotating Patterns in R d Consider a reaction diffusion system (1) u t (x,t) = A u(x,t)+f(u(x,t)), t > 0, x R d, d 2, u(x,0) = u 0 (x), t = 0, x R d. where u : R d [0, [ R N, A R N,N, f C 2 (R N,R N ). Assume a rotating wave solution u : R d [0, [ R N of (1) u (x,t) = v (e ts x) v : R d R N profile (pattern), 0 S R d,d skew-symmetric. Transformation (into a rotating frame): v(x,t) = u(e ts x,t) solves (2) v t (x,t) = A v(x,t)+ Sx, v(x,t) +f(v(x,t)), t > 0, x R d, d 2, v(x,0) = u 0 (x), t = 0, x R d. Sx, v(x) := d d S ij x j D i v(x) (drift term). i=1 j=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

6 Rotating Patterns in R d Consider a reaction diffusion system (1) u t (x,t) = A u(x,t)+f(u(x,t)), t > 0, x R d, d 2, u(x,0) = u 0 (x), t = 0, x R d. where u : R d [0, [ R N, A R N,N, f C 2 (R N,R N ). Assume a rotating wave solution u : R d [0, [ R N of (1) u (x,t) = v (e ts x) v : R d R N profile (pattern), 0 S R d,d skew-symmetric. Transformation (into a rotating frame): v(x,t) = u(e ts x,t) solves (2) v t (x,t) = A v(x,t)+ Sx, v(x,t) +f(v(x,t)), t > 0, x R d, d 2, v(x,0) = u 0 (x), t = 0, x R d. Sx, v(x) := d 1 d i=1 j=i+1 S ij (x j D i x i D j )v(x) (rotational term). Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

7 Rotating Patterns in R d Consider a reaction diffusion system (1) u t (x,t) = A u(x,t)+f(u(x,t)), t > 0, x R d, d 2, u(x,0) = u 0 (x), t = 0, x R d. where u : R d [0, [ R N, A R N,N, f C 2 (R N,R N ). Assume a rotating wave solution u : R d [0, [ R N of (1) u (x,t) = v (e ts x) v : R d R N profile (pattern), 0 S R d,d skew-symmetric. Transformation (into a rotating frame): v(x,t) = u(e ts x,t) solves (2) v t (x,t) = A v(x,t)+ Sx, v(x,t) +f(v(x,t)), t > 0, x R d, d 2, v(x,0) = u 0 (x), t = 0, x R d. v is a stationary solution of (2). Question: How to show exponential decay of v at x =? Consequence: Exponentially small error by restriction to bounded domain. Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

8 Rotating Patterns in R d Consider a reaction diffusion system (1) u t (x,t) = A u(x,t)+f(u(x,t)), t > 0, x R d, d 2, u(x,0) = u 0 (x), t = 0, x R d. where u : R d [0, [ R N, A R N,N, f C 2 (R N,R N ). Assume a rotating wave solution u : R d [0, [ R N of (1) u (x,t) = v (e ts x) v : R d R N profile (pattern), 0 S R d,d skew-symmetric. Transformation (into a rotating frame): v(x,t) = u(e ts x,t) solves (2) v t (x,t) = A v(x,t)+ Sx, v(x,t) +f(v(x,t)), t > 0, x R d, d 2, v(x,0) = u 0 (x), t = 0, x R d. v is a stationary solution of (2). d = 2: Spectral stability implies nonlinear stability. [BL] W.-J. Beyn, J. Lorenz Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

9 Example Consider the quintic complex Ginzburg-Landau equation (QCGL): ( u t = α u +u µ+β u 2 +γ u 4), u = u(x,t) C with u : R d [0, [ C, d {2,3}. For the parameters α = i, β = 5 2 +i, γ = i, µ = 1 2 this equation exhibits so called spinning soliton solutions. Applications: superconductivity, superfluidity, nonlinear optical systems. Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

10 Example Consider the quintic complex Ginzburg-Landau equation (QCGL): ( u t = α u +u µ+β u 2 +γ u 4), u = u(x,t) C with u : R d [0, [ C, d {2,3}. For the parameters α = i, β = 5 2 +i, γ = i, µ = 1 2 this equation exhibits so called spinning soliton solutions. [CMM] L.-C. Crasovan, B.A. Malomed, D. Mihalache Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

11 Example Consider the quintic complex Ginzburg-Landau equation (QCGL): ( u t = α u +u µ+β u 2 +γ u 4), u = u(x,t) C with u : R d [0, [ C, d {2,3}. For the parameters α = i, β = 5 2 +i, γ = i, µ = 1 2 this equation exhibits so called spinning soliton solutions. Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

12 Main result: Exponential decay of v Theorem: (Exponential Decay of v ) Let f(v ) = 0 and Reσ(Df(v )) < 0. Under further assumptions holds: For every 1 < p <, 0 < ϑ < 1 and for every radially nondecreasing weight function θ C(R d,r) of exponential growth rate η 0 with 0 η 2 ϑ 2 3 a 0 b 0 a 2 max p2 there exists K 1 = K 1 (A,f,v,d,p,θ,ϑ) > 0 with the following property: Every classical solution v of A v(x)+ Sx, v(x) +f(v(x)) = 0, x R d, such that v v L p (R d,r N ) and sup x R 0 v (x) v K 1 for some R 0 > 0 satisfies v v W 1,p θ (R d,r N ) (weighted Sobolev space). Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

13 Main result: The assumptions Reλ > 0 and Imλ p 2 2 p 1Reλ λ σ(a) Imλ 2 < p < 1 < p < 2 ϕ = arctan ( ) 2 p 1 p 2 Reλ A,Df(v ) R N,N simultaneously diagonalizable over C a 0 Reλ, λ a max λ σ(a) Reµ b 0 < 0 µ σ(df(v )) Σ p Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

14 Main result: The assumptions A positive function θ C(R d,r) is called a weight function of exponential growth rate η 0 provided that C θ > 0 : θ(x +y) C θ θ(x)e η y x,y R d. [ZM] S. Zelik, A. Mielke Examples: µ R, x R d ( θ 1 (x) = exp( µ x ), θ 3 (x) = exp µ ) x 2 +1, ( θ 2 (x) = cosh(µ x ), θ 4 (x) = cosh µ ) x Exponentially weighted Sobolev spaces: 1 p, k N 0 L p θ (Rd,R N ) := { v L 1 loc (Rd,R N ) θv L p < }, W k,p θ (R d,r N ) := { v L p θ (Rd,R N ) D β u L p θ (Rd,R N ) β k }. Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

15 Outline of proof: Exponential Decay of v Consider the nonlinear problem A v (x)+ Sx, v (x) +f(v (x)) = 0, x R d, d Far-Field Linearization: f C 1, Taylor s theorem, f(v ) = 0 a(x) = 1 0 Df(v +t(v (x) v ))dt, w(x) := v (x) v A w(x)+ Sx, w(x) +a(x)w(x) = 0, x R d. Q(x) Q c(x) K 1 Q ε(x) R 0 x = R Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

16 Outline of proof: Exponential Decay of v Consider the nonlinear problem A v (x)+ Sx, v (x) +f(v (x)) = 0, x R d, d Decomposition of a: Df(v )+Q(x) = 1 0 Df(v +t(v (x) v ))dt, w(x) := v (x) v A w(x)+ Sx, w(x) + (Df(v )+Q(x))w(x) = 0, x R d. Q(x) Q c(x) K 1 Q ε(x) R 0 x = R Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

17 Outline of proof: Exponential Decay of v Consider the nonlinear problem A v (x)+ Sx, v (x) +f(v (x)) = 0, x R d, d Decomposition of a: Df(v )+Q(x) = 1 0 Df(v +t(v (x) v ))dt, w(x) := v (x) v A w(x)+ Sx, w(x) + (Df(v )+Q ε (x)+q c (x))w(x) = 0, x R d. Q(x) 3. Decomposition of Q: Q c(x) Q(x) = Q ε (x)+q c (x), K 1 Q ε(x) R 0 x = R Q,Q ε,q c L (R d,r N,N ), Q ε small, i.e. Q ε L < K 1, Q c compactly supported. Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

18 Outline of proof: Exponential Decay of v Consider the nonlinear problem A v (x)+ Sx, v (x) +f(v (x)) = 0, x R d, d Decomposition of a: Df(v )+Q(x) = 1 0 Df(v +t(v (x) v ))dt, w(x) := v (x) v A w(x)+ Sx, w(x) + (Df(v )+Q ε (x)+q c (x))w(x) = 0, x R d. Q(x) 3. Decomposition of Q: Q c(x) Q(x) = Q ε (x)+q c (x), K 1 Q ε(x) R 0 x = R Q,Q ε,q c L (R d,r N,N ), Q ε small, i.e. Q ε L < K 1, Q c compactly supported. Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

19 Outline of proof: Exponential Decay of v Consider the nonlinear problem A v (x)+ Sx, v (x) +f(v (x)) = 0, x R d, d Decomposition of a: Df(v )+Q(x) = 1 0 Df(v +t(v (x) v ))dt, w(x) := v (x) v A w(x)+ Sx, w(x) + (Df(v )+Q ε (x)+q c (x))w(x) = 0, x R d. Q(x) 3. Decomposition of Q: Q c(x) Q(x) = Q ε (x)+q c (x), K 1 Q ε(x) R 0 x = R Q,Q ε,q c L (R d,r N,N ), Q ε small, i.e. Q ε L < K 1, Q c compactly supported. Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

20 Exponential Decay: To show exponential decay for the solution v of A v(x)+ Sx, v(x) +f(v(x)) = 0, x R d, d 2 investigate the far-field linearization (w.l.o.g. v = 0) A v(x)+ Sx, v(x) +(Df(v )+Q ε (x)+q c (x))v(x) = 0, x R d, d 2. Operators: Study the following operators L Q v :=A v + S, v +Df(v )v +Q ε v +Q c v, L Qε v :=A v + S, v +Df(v )v +Q ε v, L v :=A v + S, v +Df(v )v, L 0 v :=A v + S, v (Ornstein-Uhlenbeck operator). (max. domain) Ornstein-Uhlenbeck Operator Let P,B R d,d, P = P T, P > 0 and B 0. d d d d T P v(x)+ Bx, v(x) = D i (P ij D j v(x))+ D i v(x)b ij x j, x R d Here: P = I d and B = S. i=1 j=1 i=1 j=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

21 Exponential Decay: To show exponential decay for the solution v of A v(x)+ Sx, v(x) +f(v(x)) = 0, x R d, d 2 investigate the far-field linearization (w.l.o.g. v = 0) A v(x)+ Sx, v(x) +(Df(v )+Q ε (x)+q c (x))v(x) = 0, x R d, d 2. Operators: Study the following operators L Q v :=A v + S, v +Df(v )v +Q ε v +Q c v, L Qε v :=A v + S, v +Df(v )v +Q ε v, L v :=A v + S, v +Df(v )v, L 0 v :=A v + S, v (Ornstein-Uhlenbeck operator). (max. domain) Ornstein-Uhlenbeck Operator Let P,B R d,d, P = P T, P > 0 and B 0. d d d d T P v(x)+ Bx, v(x) = D i (P ij D j v(x))+ D i v(x)b ij x j, x R d Here: P = I d and B = S. i=1 j=1 i=1 j=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

22 Exponential Decay: To show exponential decay for the solution v of A v(x)+ Sx, v(x) +f(v(x)) = 0, x R d, d 2 investigate the far-field linearization (w.l.o.g. v = 0) A v(x)+ Sx, v(x) +(Df(v )+Q ε (x)+q c (x))v(x) = 0, x R d, d 2. Operators: Study the following operators L Q v :=A v + S, v +Df(v )v +Q ε v +Q c v, L Qε v :=A v + S, v +Df(v )v +Q ε v, L v :=A v + S, v +Df(v )v, (exp. decay) L 0 v :=A v + S, v (Ornstein-Uhlenbeck operator). (max. domain) Ornstein-Uhlenbeck Operator Let P,B R d,d, P = P T, P > 0 and B 0. d d d d T P v(x)+ Bx, v(x) = D i (P ij D j v(x))+ D i v(x)b ij x j, x R d Here: P = I d and B = S. i=1 j=1 i=1 j=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

23 Exponential Decay: To show exponential decay for the solution v of A v(x)+ Sx, v(x) +f(v(x)) = 0, x R d, d 2 investigate the far-field linearization (w.l.o.g. v = 0) A v(x)+ Sx, v(x) +(Df(v )+Q ε (x)+q c (x))v(x) = 0, x R d, d 2. Operators: Study the following operators L Q v :=A v + S, v +Df(v )v +Q ε v +Q c v, L Qε v :=A v + S, v +Df(v )v +Q ε v, (exp. decay) L v :=A v + S, v +Df(v )v, (exp. decay) L 0 v :=A v + S, v (Ornstein-Uhlenbeck operator). (max. domain) Ornstein-Uhlenbeck Operator Let P,B R d,d, P = P T, P > 0 and B 0. d d d d T P v(x)+ Bx, v(x) = D i (P ij D j v(x))+ D i v(x)b ij x j, x R d Here: P = I d and B = S. i=1 j=1 i=1 j=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

24 Exponential Decay: To show exponential decay for the solution v of A v(x)+ Sx, v(x) +f(v(x)) = 0, x R d, d 2 investigate the far-field linearization (w.l.o.g. v = 0) A v(x)+ Sx, v(x) +(Df(v )+Q ε (x)+q c (x))v(x) = 0, x R d, d 2. Operators: Study the following operators L Q v :=A v + S, v +Df(v )v +Q ε v +Q c v, (exp. decay) L Qε v :=A v + S, v +Df(v )v +Q ε v, (exp. decay) L v :=A v + S, v +Df(v )v, (exp. decay) L 0 v :=A v + S, v (Ornstein-Uhlenbeck operator). (max. domain) Ornstein-Uhlenbeck Operator Let P,B R d,d, P = P T, P > 0 and B 0. d d d d T P v(x)+ Bx, v(x) = D i (P ij D j v(x))+ D i v(x)b ij x j, x R d Here: P = I d and B = S. i=1 j=1 i=1 j=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

25 Exponential Decay: To show exponential decay for the solution v of A v(x)+ Sx, v(x) +f(v(x)) = 0, x R d, d 2 investigate the far-field linearization (w.l.o.g. v = 0) A v(x)+ Sx, v(x) +(Df(v )+Q ε (x)+q c (x))v(x) = 0, x R d, d 2. Operators: Study the following operators L Q v :=A v + S, v +Df(v )v +Q ε v +Q c v, (exp. decay) L Qε v :=A v + S, v +Df(v )v +Q ε v, (exp. decay) L v :=A v + S, v +Df(v )v, (exp. decay) L 0 v :=A v + S, v (Ornstein-Uhlenbeck operator). (max. domain) Ornstein-Uhlenbeck Operator Let P,B R d,d, P = P T, P > 0 and B 0. d d d d T P v(x)+ Bx, v(x) = D i (P ij D j v(x))+ D i v(x)b ij x j, x R d Here: P = I d and B = S. i=1 j=1 i=1 j=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

26 Exponential Decay: To show exponential decay for the solution v of A v(x)+ Sx, v(x) +f(v(x)) = 0, x R d, d 2 investigate the far-field linearization (w.l.o.g. v = 0) A v(x)+ Sx, v(x) +(Df(v )+Q ε (x)+q c (x))v(x) = 0, x R d, d 2. Operators: Study the following operators L Q v :=A v + S, v +Df(v )v +Q ε v +Q c v, (exp. decay) L Qε v :=A v + S, v +Df(v )v +Q ε v, (exp. decay) L v :=A v + S, v +Df(v )v, (exp. decay) L 0 v :=A v + S, v (Ornstein-Uhlenbeck operator). (max. domain) [MPV] G. Metafune, D. Pallara, V. Vespri [MPRS] G. Metafune Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

27 The operator L 0 Ornstein-Uhlenbeck operator [L 0v](x) = A v(x)+ Sx, v(x), x R d, d 2. ( Heat kernel H 0(x,ξ,t) = (4πtA) d 2 exp (4tA) 1 e ts x ξ 2), x,ξ R d, t > 0. semigroup theory ւ unique solv. of resolvent equ., 1 p < (λi A p)v = g L p. Semigroup in L p (R d,r N ), 1 p [T 0(t)v](x) = H 0(x,ξ,t)v(ξ)dξ, t > 0. R d strong continuity Infinitesimal generator (A p,d(a p)), 1 p <. A-priori estimates exponential decay, 1 p < v W 1,p ց L 0: L p -resolvent est. max. domain and max. realization, 1 < p < A p = L 0 on D(A p) = D p (L 0). θ. Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

28 Spectrum of localized rotating waves Linearization about the profile of the rotating wave [Lv](x) = A v(x)+ Sx, v(x) +Df(v (x))v(x), x R d, d 2. Eigenvalue problem [Lv](x) = λv(x), x R d, d 2, λ C. A rotating wave solution u (x,t) = v ( e ts x ) is spectrally stable if Decompose the spectrum σ(l) into σ(l) {λ C Reλ 0}. σ(l) = σ ess (L) σ pt (L), with essential spectrum σ ess (L) and point spectrum σ pt (L). Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

29 Illustration: Point spectrum of L λ (σ(s) {λ+µ λ,µ σ(s), λ µ}) σ pt (L) with algebraic multiplicity Imλ Imλ 1 Imλ i(σ 1 +σ 2 ) 1 Imλ i(σ 1 +σ 2 ) 1 iσ 1 2 iσ 1 1 i(σ 1 σ 2 ) 1 i(σ 1 σ 2 ) 1 iσ 1 2 iσ 1 1 iσ 2 2 iσ Reλ iσ Reλ iσ Reλ iσ Reλ iσ 2 1 i(σ 1 σ 2 ) 1 i(σ 1 σ 2 ) 1 iσ 1 2 iσ 1 1 i(σ 1 +σ 2 ) 1 i(σ 1 +σ 2 ) d = 2 dimse(2) = 3 d = 3 dimse(3) = 6 d = 4 dimse(4) = 10 d = 5 dimse(5) = 15 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

30 Point spectrum of L Theorem: (Point spectrum of L) Let A R N,N satisfy the main assumptions, f C 2 (R N,R N ), 1 < p < and let v be a classical solution of [L 0 v](x)+f(v(x)) = 0. Then v(x) = C rot x +I d C tra, v (x), x R d, C rot so(d), C tra R d solves Lv = λv, whenever (λ,(c rot,c tra )) solves In particular: λc rot = SC rot + ( SC rot) T, λc tra = SC tra. σ(s) {λ+µ λ,µ σ(s), λ µ} σ pt (L), v(x) = Sx, v (x) eigenfunction of λ = 0 for every d 2, group action σ pt (L), Theorem also valid for spiral waves, scroll waves, scroll rings. Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

31 Point spectrum of L Theorem: (Point spectrum of L) Let A R N,N satisfy the main assumptions, f C 2 (R N,R N ), 1 < p < and let v be a classical solution of [L 0 v](x)+f(v(x)) = 0. Then v(x) = C rot x +I d C tra, v (x), x R d, C rot so(d), C tra R d solves Lv = λv, whenever (λ,(c rot,c tra )) solves In particular: λc rot = SC rot + ( SC rot) T, λc tra = SC tra. σ(s) {λ+µ λ,µ σ(s), λ µ} σ pt (L), v(x) = Sx, v (x) eigenfunction of λ = 0 for every d 2, group action σ pt (L), Theorem also valid for spiral waves, scroll waves, scroll rings. Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

32 Exponential decay of v Theorem: (Exponential decay of v) Let the assumptions of the main result be satisfied. The every classical solution v L p (R d,c N ) of A v(x)+ Sx, v(x) +Df(v (x))v(x) = λv(x), x R d, with λ C and Reλ b0 3 satisfies v W 1,p θ (R d,c N ) v exp. localized v exp. localized (with same rate) Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

33 Outline of proof: Point spectrum of L (3) 0 = A v (x)+ Sx, v (x) +f(v (x)), x R d, d Group action: Apply a(r,τ) to (3) 0 = a(r,τ)(a v (x)+ Sx, v (x) +f(v (x))) 2. Derivative d d(r,τ) at (R,τ) = (I d,0) leads to d(d+1) 2 equations 0 =(x j D i x i D j )(A v (x)+ Sx, v (x) +f(v (x))) 0 =D l (A v (x)+ Sx, v (x) +f(v (x))) for i = 1,...,d 1, j = i +1,...,d, l = 1,...,d. 3. Commutator relations for differential operators yield, D (ij) := x j D i x i D j ( ) 0 =L D (ij) v (x) + 0 =L(D l v (x)) d S in D (jn) v (x) n=1 n j d S ln D n v (x) n=1 d S jn D (in) v (x) n=1 n i Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

34 Outline of proof: Point spectrum of L (3) 0 = A v (x)+ Sx, v (x) +f(v (x)), x R d, d Group action: Apply a(r,τ) to (3) 0 = a(r,τ)(a v (x)+ Sx, v (x) +f(v (x))) 2. Derivative d d(r,τ) at (R,τ) = (I d,0) leads to d(d+1) 2 equations 0 =(x j D i x i D j )(A v (x)+ Sx, v (x) +f(v (x))) 0 =D l (A v (x)+ Sx, v (x) +f(v (x))) for i = 1,...,d 1, j = i +1,...,d, l = 1,...,d. 3. Commutator relations for differential operators yield, D (ij) := x j D i x i D j ( ) 0 =L D (ij) v (x) + 0 =L(D l v (x)) d S in D (jn) v (x) n=1 n j d S ln D n v (x) n=1 d S jn D (in) v (x) n=1 n i Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

35 Outline of proof: Point spectrum of L (3) 0 = A v (x)+ Sx, v (x) +f(v (x)), x R d, d Group action: Apply a(r,τ) to (3) 0 = a(r,τ)(a v (x)+ Sx, v (x) +f(v (x))) 2. Derivative d d(r,τ) at (R,τ) = (I d,0) leads to d(d+1) 2 equations 0 =(x j D i x i D j )(A v (x)+ Sx, v (x) +f(v (x))) 0 =D l (A v (x)+ Sx, v (x) +f(v (x))) for i = 1,...,d 1, j = i +1,...,d, l = 1,...,d. 3. Commutator relations for differential operators yield, D (ij) := x j D i x i D j ( ) 0 =L D (ij) v (x) + 0 =L(D l v (x)) d S in D (jn) v (x) n=1 n j d S ln D n v (x) n=1 d S jn D (in) v (x) n=1 n i Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

36 Outline of proof: Point spectrum of L 3. Commutator relations for differential operators yield, D (ij) := x j D i x i D j ( ) 0 =L D (ij) v (x) + 0 =L(D l v (x)) d S in D (jn) v (x) n=1 n j d S ln D n v (x) n=1 d S jn D (in) v (x) n=1 n i 4. Finite-dimensional eigenvalue problem: The ansatz v(x) = d 1 d i=1 j=i+1 reduces Lv = λv to a Note: S is unitary diagonalizable. C rot ij (x j D i x i D j )v (x)+ d l=1 λc rot = SC rot + ( SC rot) T, λc tra = SC tra. Cl tra D l v (x), Cij rot,cl tra C Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

37 Outline of proof: Point spectrum of L 3. Commutator relations for differential operators yield, D (ij) := x j D i x i D j ( ) 0 =L D (ij) v (x) + 0 =L(D l v (x)) d S in D (jn) v (x) n=1 n j d S ln D n v (x) n=1 d S jn D (in) v (x) n=1 n i 4. Finite-dimensional eigenvalue problem: The ansatz v(x) = d 1 d i=1 j=i+1 reduces Lv = λv to a Note: S is unitary diagonalizable. C rot ij (x j D i x i D j )v (x)+ d l=1 λc rot = SC rot + ( SC rot) T, λc tra = SC tra. Cl tra D l v (x), Cij rot,cl tra C Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

38 Illustration: Essential spectrum of L ( λ(ω) + i k X l=1 2 nl σl λ(ω) eigenvalue of ω A Df (v ) d = 2 or 3 Denny Otten (Bielefeld University) d = 4 (not dense) Spatial decay of rotating waves in parabolic systems ) σess (L) d = 4 (dense) Bielefeld / 22

39 Essential spectrum of L Theorem: (Essential spectrum of v) Let the assumptions of the main result be satisfied. Moreover, let ±iσ 1,...,±iσ k denote the nonzero eigenvalues of S and let λ(ω) denote an eigenvalue of ω 2 A Df(v ) for some ω R. Then { in L p (R d,c N ). λ = λ(ω) i } k n l σ l C n l Z, ω R σ ess (L) l=1 Far-field linearization σ ess (L) Dispersion relation: λ σ ess (L) if ( ) k det λi N +ω 2 A+i n l σ l I N Df(v ) = 0 for some κ R. l=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

40 Outline of proof: Essential spectrum of L Linearization at the profile v : [Lv](x) = A v(x)+ Sx, v(x) +Df(v )v(x)+q(x)v(x) Q(x) := Df(v (x)) Df(v ), sup Q(x) 2 0 as R x R 1. Orthogonal transformation: S R d,d, S T = S, S = PΛ S block PT. T 1 (x) = Px yields with [L 1 v](x) = A v(x)+ Λ S block x, v(x) +Df(v )v(x)+q(t 1 (x))v(x) Λ S block x, v(x) = k σ l (x 2l D 2l 1 x 2l 1 D 2l )v(x). l=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

41 Outline of proof: Essential spectrum of L Orthogonal transformation: [L 1 v](x) = A v(x)+ Λ S block x, v(x) +Df(v )v(x)+q(t 1 (x))v(x) Λ S block x, v(x) = k σ l (x 2l D 2l 1 x 2l 1 D 2l )v(x) l=1 2. Several planar polar coordinates: Transformation ( ) ( ) x2l 1 rl cosφ = T(r x l,φ l ) := l, l = 1,...,k, φ 2l r l sinφ l ] π,π], r l > 0. l yields for ξ = (r 1,φ 1,...,r k,φ k,x 2k+1,...,x d ) with total transformation T 2 (ξ), Q(ξ) := Q(T 1 (T 2 (ξ))) [L 2 v](x) =A [ k l=1 ( 2 r l + 1 r l rl + 1 rl 2 2 φ l )+ d l=2k+1 2 x l ]v(ξ) k σ l φl v(ξ)+df(v )v(ξ)+q(ξ)v(ξ), l=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

42 Outline of proof: Essential spectrum of L Several planar polar coordinates: [L 2 v](ξ) =A [ k l=1 ( 2 r l + 1 r l rl + 1 rl 2 2 φ l )+ d l=2k+1 2 x l ]v(ξ) k σ l φl v(ξ)+df(v )v(ξ)+q(ξ)v(ξ), l=1 ξ = (r 1,φ 1,...,r k,φ k,x 2k+1,...,x d ), Q(ξ) := Q(T 1 (T 2 (ξ))) 3. Simplified operator (far-field linearization): Neglecting O ( 1 r) terms yields [ L sim 2 v ] [ k (x) = A r 2 l + l=1 d l=2k+1 2 x l ]v(ξ) k σ l φl v(ξ)+df(v )v(ξ). l=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

43 Outline of proof: Essential spectrum of L Simplified operator (far-field linearization): [ L sim 2 v ] [ k (ξ) = A r 2 l + l=1 d l=2k+1 2 x l ]v(ξ) k σ l φl v(ξ)+df(v )v(ξ) l=1 4. Angular Fourier decomposition: ( ( k k v(ξ) = exp iω r l )exp i n l φ l )ˆv,n l Z, ω R, ˆv C N, ˆv = 1 l=1 l=1 φ l ] π,π], r l > 0, l = 1,...,k, yields [( λi L sim 2 ) v ] (ξ) = (λi N +ω 2 A+i ) k n l σ l I N Df(v ) v(ξ). l=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

44 Outline of proof: Essential spectrum of L Angular Fourier decomposition: [( λi L sim 2 ) v ] (ξ) = (λi N +κ 2 A+i ) k n l σ l I N Df(v ) v(ξ). l=1 n l Z, κ R, ±iσ l nonzero eigenvalues of S R d,d 5. Finite-dimensional eigenvalue problem: [( ) ] λi L sim 2 v (ξ) = 0 for every ξ if λ C satisfies ( ω 2 A Df(v ) )ˆv ( k = λ+i n l σ l )ˆv, for some ω R. l=1 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

45 Example Consider the quintic complex Ginzburg-Landau equation (QCGL): ( u t = α u +u µ+β u 2 +γ u 4), u = u(x,t) C with u : R d [0, [ C, d {2,3}. For the parameters α = i, β = 5 2 +i, γ = i, µ = 1 2 this equation exhibits so called spinning soliton solutions σess approx σ approx pt σ ess(l) σ(s) {0} Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

46 Figure: Eigenfunctions of QCGL for a spinning soliton with d = 3 Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

47 Work in progress exponential decay in space of continuous functions rotating waves in bounded domains approximation theorem for rotating waves asymptotic boundary conditions numerical computations (interaction of multisolitons) Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22

48 Work in progress exponential decay in space of continuous functions rotating waves in bounded domains approximation theorem for rotating waves asymptotic boundary conditions numerical computations (interaction of multisolitons) Denny Otten (Bielefeld University) Spatial decay of rotating waves in parabolic systems Bielefeld / 22