MS: Nonlinear Wave Propagation in Singular Perturbed Systems

Transcription

1 MS: Nonlinear Wave Propagation in Singular Perturbed Systems P. van Heijster: Existence & stability of 2D localized structures in a 3-component model. Y. Nishiura: Rotational motion of traveling spots in dissipative systems. M. Wechselberger: A geometric twist on tactically-driven cell migration. P. Zegeling: Instability of travelling waves in a 2D nonequilibrium Richard s equation.

2 MS: Nonlinear Wave Propagation in Singular Perturbed Systems Traveling spots, fronts etc. I II Different spatial scales: diffusion coefficients or reaction terms Gray-Scott, Schnakenberg, Gierer-Meinhardt III IV Experiment u v w + t 0 h(u, w) g(u, w)v Modelling ARD model = x h(u, w) 0 f(u, w) + ε u v w xx Geometric Theory PDE Theory A Rankine-Hugoniot + Lax condition L S r Sa on otential propagation

3 MS: Nonlinear Wave Propagation in Singular Perturbed Systems Traveling spots, fronts etc. Different spatial scales: diffusion coefficients or reaction terms Gray-Scott, Schnakenberg, Gierer-Meinhardt I II Experiment III IV u v w + t 0 h(u, w) g(u, w)v Modelling ARD model = x h(u, w) 0 f(u, w) + ε u v w xx Geometric Theory PDE Theory A Rankine-Hugoniot + Lax condition L S r Sa

4 MS: Nonlinear Wave Propagation in Singular Perturbed Systems Traveling spots, fronts etc. Different spatial scales: diffusion coefficients or reaction terms Gray-Scott, Schnakenberg, Gierer-Meinhardt Experiment u v w + t 0 h(u, w) g(u, w)v Modelling ARD model = x h(u, w) 0 f(u, w) + ε u v w xx Geometric Theory PDE Theory A Rankine-Hugoniot + Lax condition L S r Sa

5 MS: Nonlinear Wave Propagation in Singular Perturbed Systems Traveling spots, fronts etc. Different spatial scales: diffusion coefficients or reaction terms Gray-Scott, Schnakenberg, Gierer-Meinhardt Experiment u v w t + 0 h(u, w) g(u, w)v Modelling ARD model x = h(u, w) 0 f(u, w) + ε u v w xx Geometric Theory PDE Theory A Rankine-Hugoniot + Lax condition L S r Sa on otential propagation

6 MS: Nonlinear Wave Propagation in Singular Perturbed Systems Traveling spots, fronts etc. Different spatial scales: diffusion coefficients or reaction terms Gray-Scott, Schnakenberg, Gierer-Meinhardt

7 Existence & stability of 2D localized structures in a 3-component model Peter van Heijster DSPDES 10 MS: Nonlinear Wave Propagation in Singular Perturbed Systems Barcelona, Spain, June 2010 Collaborator: B. Sandstede Previous collaborators (1D): A. Doelman, T.J Kaper, K. Promislow, Research funded by:

8 Outline Introduction Spot - Existence - Stability Future work

9 Paradigm system Generalized FitzHugh-Nagumo Equation: where 0 < ε << 1; D>1; 0<τ,θ; α,β,γ are constants. Physical background: Gas-discharge experiments by Purwins et al. Inspiration: Numerical collision experiments by Nishiura et al. Motivation: Rich behavior and transparent structure enables rigorous mathematical analysis.

10 Paradigm system Generalized FitzHugh-Nagumo Equation: where 0 < ε << 1; D>1; 0<τ,θ; α,β,γ are constants. Physical background: Gas-discharge experiments by Purwins et al. Inspiration: Numerical collision experiments by Nishiura et al. Motivation: Rich behavior and transparent structure enables rigorous mathematical analysis. Goal: Understanding radially symmetric stationary spots Influence of the third component

11 Experiments Set up: Observed patterns: I II III IV

12 Radially symmetric spot 1D: 1-pulse

13 Radially symmetric spot 1D: 1-pulse

14 Radially symmetric spot 2D: Spot

15 More complex structures Ring Spot-ring Both unstable

16 Spot: existence Stationary: Radially symmetric: ODE: B.C.: Neumann at the core and (background)

17 Spot: existence Stationary: Radially symmetric: ODE: B.C.: Neumann at the core and (background) 1D problem with singularity at the core r=0.

18 Spot: existence Stationary: Radially symmetric: ODE: B.C.: Neumann at the core and (background) 1D problem with singularity at the core r=0. 1D result [Doelman, H., Kaper]: There exist a stationary 1-pulse solution with width xp if there exists a xp solving: where v0, w0 are given by

19 Schematic 1D-spot 4 different regions Core : r=0; and Slow : Fast : around r = R1, v,w= constant Slow :, asymptotes to Idea: Use these properties to construct a spot.

20 Flavor: fast field Rescale: ODE:, s fast coordinate

21 Flavor: fast field Rescale: ODE:, s fast coordinate Fast reduced system: ε =0 Centered around R1: and R1 = O(1) wrt r.

22 Flavor: fast field Rescale: ODE:, s fast coordinate Fast reduced system: ε =0 Centered around R1: and R1 = O(1) wrt r. Hamiltonian: Heteroclinic connection:

23 ε=0: 4-dim. invariant manifolds: Flavor: Fenichel ε 0: 4-dim. locally invariant manifolds Also unstable and stable manifolds of persist

24 ε=0: 4-dim. invariant manifolds: Flavor: Fenichel ε 0: 4-dim. locally invariant manifolds Also unstable and stable manifolds of persist

25 ε=0: 4-dim. invariant manifolds: Flavor: Fenichel ε 0: 4-dim. locally invariant manifolds Also unstable and stable manifolds of persist u

26 Flavor: slow field I : u=1 (to leading order) ODE: Modified Bessel functions:

27 Flavor: slow field I : u=1 (to leading order) ODE: Modified Bessel functions: 8 6 v 4 2 w

28 Flavor: slow field II Similar on : u=-1 Bounded at infinity Modified Bessel functions:

29 Flavor: slow field II Similar on : u=-1 Bounded at infinity Modified Bessel functions: v w

30 Flavor: core Scale: Matches smoothly with slow field Neumann at core:

31 Flavor: core Scale: Matches smoothly with slow field Neumann at core:

32 Flavor: core Scale: Matches smoothly with slow field Neumann at core: 8 6 v 4 2 w

33 Flavor: matching Slow components do not change over fast field: Match in fast field to determine four constants: Gives (using a Wronskian identity)

34 Flavor: matching Slow components do not change over fast field: Match in fast field to determine four constants: Gives (using a Wronskian identity) w v Slow components in fast field

35 Flavor: Radius Unperturbed fast system was Hamiltonian: In perturbed system: Hamiltonian on. Therefore (Melnikov integral), This yields:

36 Flavor: Radius Unperturbed fast system was Hamiltonian: In perturbed system: Hamiltonian on. Therefore (Melnikov integral), This yields: Theorem: There exists a stationary radially symmetric spot solution with radius R1 if there exists an R1 solving:

37 Spot: stability (U,V,W)(r,φ,t) = (us,v s,w s )(r) + (u,v,w)(r)e λt+iφl ODE:

38 Spot: stability (U,V,W)(r,φ,t) = (us,v s,w s )(r) + (u,v,w)(r)e λt+iφl ODE: Essential spectrum: left half plane Eigenvalues: Note: α,β<0: Spot is unstable wrt l=0 (translation invariance in y-direction)

39 Spectrum Eigenvectors for l=0 and l=5: Again 4 regions Core Ic : Neumann and u=0 Slow : Fast If : v,w constant Same type of analysis - 1.0

40 Spectrum Eigenvectors for l=0 and l=5: Again 4 regions Core Ic : Neumann and u=0 Slow : Fast If : v,w constant Same type of analysis α,β>0: l= Υ l=2 α,β<0: l= Υ l=2

41 Future work: interaction U Stationary Solution Δ Γ Initial Condition Question: Given the system-parameters and an initial condition, can we quantitatively predict how the structure evolves in time? ξ

42 Future work: interaction U Stationary Solution Δ Γ Initial Condition Question: Given the system-parameters and an initial condition, can we quantitatively predict how the structure evolves in time? ξ Answer: For 1D, yes! We can derive a system of ODEs describing the motion of the separate fronts. 1-Pulse:

43 2D: interaction Can we derive something similar for planar structures? Spot-ring

44 2D: interaction 3 interacting spots Renormalization group method used for the 1D problem does not work in higher dimensions

45 Future: traveling spot Traveling spot (and growing) No reduction to an ODE problem.

46 Thanks! Questions?