Fully discrete traveling waves from semi discrete traveling waves

Fully discrete traveling waves from semi-discrete traveling waves S. Benzoni-Gavage May 20th, 2003 The purpose of this note is to state and sketch the proof of Theorem B in [4]. For the reader's convenience we adopt the same notations as Chow, Mallet-Paret and Shen. Their result concerns a general Lattice Dynamical System (LDS) x˙ = F (x) , (1) where F is a smooth function in X = ∞(Z,Rd) that commutes with the shift operator, S : x 7? Sx ; (Sx)j = xj?1 , and the fully discrete counterpart of (1) obtained by Euler discretization xn+1 = xn + hF (xn) . (2) This is called a Coupled Map Lattice, associated with the map Gh : x 7? Gh(x) := x + hF (x) . The result of Chow, Mallet-Paret and Shen reported here shows that spectrally stable traveling wave solutions to (1) give rise to traveling wave solutions to (2) for small enough h. Their spectral stability requirement needs some explanation. Assume that x = p(t) is a traveling wave solution of (1), of positive speed c, i. e. pj(t) = ?(j ? c t) for every j ? Z and t ? R.

• stable traveling wave


• original coordinates


• tz


• invariant manifold


• spectrally stable


• manifold


• spectral assumption


• cr map


• positive h0


• traveling wave