4 1. INTRODUCTION

• Modulation equations near sideband instabilities (§7):

When the underlying wave trains become sideband unstable, the Burgers

equation does no longer provide an accurate description of the dynamics of

slow modulations. Instead, depending on the form of the linear dispersion

relation, it is the Korteweg–de Vries or the Kuramoto–Sivashinsky equa-

tion that takes its role. We discuss their validity properties for reaction-

diffusion systems.

• Existence and stability of weak shocks (setup: §4.1; results: §4.5; proofs:

§8):

We show that the viscous shock fronts in the Burgers equation correspond

to genuine modulated waves of the underlying reaction-diffusion system.

In other words, we construct stable waves that are time-periodic in an

appropriately moving coordinate frame and whose profile converges, as

x → ±∞, to two wave trains with different, but almost identical, wave

number. The speed of these interfaces is determined by the Rankine–

Hugoniot condition with the flux given by the nonlinear dispersion relation

of the wave trains. The group velocities of the asymptotic wave trains,

computed in a frame moving with the interface, are directed toward the

interface.

• Global analysis of trains of well-separated pulses (§9):

In the limit of infinite wavelength (or zero wave number), wave trains are

made up by an infinite number of well-separated pulses whose dynamics

can be described formally by a lattice dynamical system. In this descrip-

tion, modulated fronts that connect two such wave trains can be found

as heteroclinic orbits to a certain delay equation. We show that these

heteroclinic solutions exist between any two wave trains.

• Applications (§10):

The results presented here are applied to the FitzHugh–Nagumo equation

and to the Taylor–Couette problem.

1.2. Slowly-varying modulations of nonlinear wave trains

We shall investigate the dynamics of weakly-modulated nonlinear wave trains

in partial differential equations (PDEs) on the real line. To set the scene, suppose

that we are given a reaction-diffusion system

(1.1) ∂tu = D∂xxu + f(u), x ∈ R, u ∈

Rd.

Starting point of our investigation are wave trains which are solutions u(x, t) =

u0(ωt − kx) of (1.1) that are 2π-periodic in their argument θ = ωt − kx. Thus, ω

may be interpreted as the temporal frequency of the wave train and k as its spatial

wave number; their quotient cp = ω/k gives the wave speed, or phase velocity, of the

nonlinear wave. Typically, wave trains exist for an entire

range1

of wave numbers

k, and both the profile u0(θ) and the frequency ω will depend on the choice of k.

To reflect this fact, we write the travelling wave as

(1.2) u(x, t) = u0(ωt − kx; k)

and denote the frequency ω selected by the wave number k by ωnl(k); we shall refer

to this function as the nonlinear dispersion relation.

1

See §4.1 for details.