TRAVELLING WAVES. Morteza Fotouhi Sharif Univ. of Technology

Transcription

1 TRAVELLING WAVES Morteza Fotohi Sharif Univ. of Technology Mini Math NeroScience Mini Math NeroScience Agst 28

2 REACTION DIFFUSION EQUATIONS U = DU + f ( U ) t xx x t > U n D d 1 = d j > d n 2

3 Travelling wave is a soltion of the form Uxt (,) = Vx ( ct) = x ct cv = DV + f ( V ) V V = W W D 1 = [ cw + f( V)] 3

4 REST STATE W x eqilibrim V 4

5 PULSE WAVE W c x V Homoclinic Orbit 5

6 FRONT WAVE W c x Hetroclinic Orbit V 6

7 PERIODIC WAVE W c x Periodic Orbit V 7

8 EXAMPLE FitzHgh-Nagmo g v = v + f () v t t xx = βv fv () = v ( α v )( v 1) 1), < α < 1, 1 < β V = W W = cw f( V) + U U β = V c 8

9 9

10 STABILITY If U(x) is a stationary soltion of Reaction- Diffsion eqation, U t = DU xx + f ( U ) we call it stable when for every initial vale close to U in some norm, i.e. the soltion satisfies U < ε X (., t) U( + h) X < δ Frthermore, it is called asymptotically stable, if it is stable and tends towards Ux ( + h), for a constant h. (., t) U( + h) X 1

11 STABLITY OF TRAVELLING WAVES Uxt (,) = Vx ( ctt,) V = DV + cv + f ( V ) t Vx ( ct ) Travelling wave is a stationary soltion of this PDE. We mean its (asymptotically) y) stability as a stable soltion of this PDE. 11

12 LINEAR STABILITY Linearize the eqation V = DV + cv + f ( V ) t arond the stationary soltion V() V t = DV + cv + f U ( V) V L = D + c + f U ( V) 12

13 SPECTRUM Resolvent set ρ( L) { λ L λi L : D ( L ) X X = has a bonded inverse } K > : h X! U X,( L λi) U = h U X K h X 13

14 Spec ( L ) = \ ρ ( L ) = Σ Σ pt ess Σ pt : point spectrm or eigenvale defined as the kernel of L λi is nontrivial. Σ = Spec ( L )\ Σ ess pt is essential spectrm Example: L : a a1 a a1 (,, ) (,,, ) λ = Spec( L) is not an eigenvale bt becase L = (1,,, ) doesn t have 14 soltion in

15 SPECTRUM OF LINEAR EQUATION Proposition: If () is a travelling wave soltion and then V V, Spec( L). L = D + c + f U ( V) Proof. Differentiate We fid find DV + cv + f ( V ) = LV = 15

16 NONLINEAR STABILITY If is a travelling wave soltion of U t = DU xx + f ( U ) with, is a simple eigenvale of L and the other V () λ = {Re( λ) α < } spectrm are located in, then V is asymptotically stable. 16

17 CASE1: REST STATE, ν V() = V () = e n ν, \{} L = λ Sbstitte for some in the eqation to find eigenvales. We find 2 = [ + + U ( )] λ ν D cνi f V 2 d(, λν): = det[ νd + ( cν λ) I + f ( V )] U 17

18 CASE1: REST STATE, V() = V Theorem: Spec( L) = { λ d(, ik) = Proof. λ for some k } d(, λ ik), k Assme, we will show that D + c + ( f ( V ) I ) = h( ) h U λ X for every, has soltion and X K h A( λ) 18 = + v v h X

19 A ( λ ) I = 1 1 D [ λi fu ( V)] cd A( λ) 1 det[ A( λ) νi] = d( λ, ν) det D hyperbolic since d(, λ ik), k is hyperbolic E s ( λ ) = E ( λ ) = stable sbspace, nstable sbspace E s n ( λ) E ( λ) = 19

20 STRUCTURE OF SPEC(L) Spec ( L ) = { λ d ( λ, ik ) = for some k } = { λ SpecA( λ) i } f U ( V ) Spec ( L ) d ( λ, ik ) =, d λ( λ, ik ) ( ik) k k All eigenvales of lie in If, then there λ is a crve defined for sch that λ ( ik ) = λ, λ ( ik ) SpecA ( λ ) k As, we have Reλ 2

21 STRUCTURE OF SPEC(L) Spectrm lies in sector { λ < R} {arg λ > π δ} Assme that where i e ϕ λ = SpecL 2 ε < ε 1, ϕ π δ then we show that roots of i ϕ (, ) det[ 2 ( e d λν = νd + cν ) I + f ( )] 2 U V = ε ν are far from i. Let ν = ε 2 i ϕ 2 det[ ν D + ( cνε e ) I + ε f ( V )] = U 21

22 ε = e ν =± iϕ/2 For, we have is far from d j imaginary axis. And for < ε 1, ν = ν + O () ε is far too. 22

23 EXAMPLE = + a t xx v + cv + av = 2 d (, λ ik ) = k + ick + a λ = For negative parameter a< the rest wave (x,t)= is stable. 23

24 CASE 2: PERIODIC WAVE, V( + q) = V( ) = ( L λ I ) = D + c + ( f ( V ( )) λ I ) I A(, λ) = = 1 1 v v D [ λi f ( ( ))] U V cd v U A ( + q, λ ) = A (, λ ) Floqet representation B( ( ) () (, ) λ R λ e v = v 24

25 The point spectrm is empty. Spec ( L ) = { λ det( B ( λ ) ik ) = for somek } for somek = { λ Spec( B( λ)) i } Eigenfnctions are of the form () = () e ik where ( + q) = ( ) per per Sectoriality of spectrm is also tre in this case. per 25

26 CASE 3: FRONT, V() V AS ± ± L = D + c + fu ( V ( )) A(, λ) I = 1 1 D [ λi fu ( V( ))] cd E s I lim A ( λ, ) = A ( ) ± λ = 1 1 D [ λi f ( )] U V± cd ( V +, λ ) = : ( ) 2n + v v, v v V E (, λ ) = : () as where + ( ) = 2n as, where v v ( ) = v v

27 λ is in the resolvent set of L if and only if, A± ( λ) are both hyperbolic with the same Morse index, and + dim E ( λ) = dim E ( λ) E (, λ) E s (, λ) n + = λ is in the point spectrm pt, if and only if, A± ( λ) are both hyperbolic with the same Morse index, bt + dim E ( λ) = dim E ( λ) E s + (, λ) E (, λ) 27

28 λ is in the essential spectrm ess, if either at least one of the two asymptotic matrices A ( λ) is not hyperbolic, or else if it does, bt the Morse indices are different. ± 28

29 CASE 4: PULSE, V() V AS ± Special case of front wave with this different that the Morse indices are always the same. lim A( λ, ) = A ( λ ) ± 29

30 EVANS FUNCTION Choose analytic bases s { V j ( λ )} j = 1,, kand { V ( λ )} = 1,, for s E (, λ ) and E (, λ), respectively. s s 1 k 1 n k j j n k E ( λ ) = det[ V ( λ ),, V ( λ ), V ( λ ),, V ( λ )] Reslt: I. ( ) s E λ = E (, λ) E (, λ) λis an eigenvale. II. λ The order of as a zero of the Evans fnction is eqal to the algebraic mltiplicity of as an eigenvale of L. 3 λ

31 EXAMPLE Stationary soltion: = + t xx 3 qx ( ) = 2Sechx Linear eqation: v = v + (3 q ( x ) 1) v t xx 2 L = + (3 q( x) 1) xx Ax (, λ) = A () λ 2 ± = 3() qx 1 λ 1 λ Σ ess = (, 1) 31

32 1+λ λ x λ 2 ( x, λ ) = e [1+ 1+ λ tanh( x) Sech ( x )] λ x λ 2 + ( x, λ) = e [ λ tanh( x ) Sech ( x )] 3 (, λ) (, λ) + 2 E( λ) = det = λ( λ 3) 1+ λ (, λ) + (, λ) 9 Σ pt = { 1,,3} 32

33 NEURAL FIELD: INTEGRO-DIFFERENTIAL EQUATION 1 xt (, ) α t + = xt (,) + γ wyfx ()(( yt,)) dy Rest state: xt (,) = Linear Eqation: + = γ f ( ) w ( y ) dy 1 xt (, ) α t + = xt (,) + β wyx ()( ytdy,) β = γ f ( ) 33

34 L = + w()( y x y) dy β + Eigenfnctions: ikx x ( ) = e λ + 1 = βwk ˆ( ) If we assme that wy () = w( y), then wk ˆ( ) is a real even fnction of k and the stability condition is β wˆ < 1 max 34

35 REFERENCE B. Sandstede, Stability of travelling waves, Handbok of Dynamical Systems II, 22. A. I. Volpert, V. A. Volpert, and V. A. Volpert, Traveling wave soltions of parabolic systems, Translations of Mathematical Monographs 14, American Mathematical Society, L. Zhang S. Coombes C.R. Laing 35