The monograph gives a comprehensive collection of results on the justification of the reduction of a general system of reaction-diffusion equations to the viscous Burgers equation
\[ q_T= \alpha q_{XX}+ \beta qq_X, \]
where \(\alpha\) and \(\beta\) are numerical coefficients computed from linear and nonlinear dispersion relations. The Burgers equation describes perturbations of slowly modulated periodic wave trains
\[ u(x,t)= u_0(\omega t- kx- \Phi(X,T); k+\delta q(X,T)), \quad X= \delta (x- c_g t), \quad T= \delta^2t, \]
where \(\delta > 0\) is a small parameter, \(q(X,T)=\Phi_X(X,T)\), and \(u_0(\omega t - kx; k)\) is a periodic wave solution of the general reaction-diffusion equations
\[ u_t= D u_{xx} + f(u), \quad x \in \mathbb R, \quad u \in \mathbb R^d. \]
The following problems are investigated and reviewed in the monograph:

1)

Validity results for the Burgers equation on the time scale of order \(\delta^{-2}\). The error estimates are uniform in the spatial variable \(x\) provided \(q(X,T)\) approaches limits as \(X \to \pm \infty\).

2)

Modulation equations near sideband instabilities are modified as the Kuramoto-Sivashinsky-type equations.

3)

Existence and stability of weak shocks are investigated. These solutions correspond to stable waves that are time-periodic in a moving coordinate frames and whose profiles converge to two wave trains with different but almost identical wave number. The speed of the interfaces is determined by the Rankine-Hugoniot condition.

4)

Global analysis of trains of well-separated pulses is developed in the limit of infinite wavelength. Dynamics is described formally by a lattice dynamical system. It is shown that the heteroclinic orbits exist between any two wave trains of well-separated pulses.

Applications include the FitzHugh-Nagumo equation and the Taylor-Couette problem. The Burgers equation and the complex cubic Ginzburg-Landau equation are analyzed in full length.