  An equation to describe nearly one-dimensional traveling-waves patterns is
put forward. This is a dispersive generalization of the classical
Newell-Whitehead-Segel (NWS) equation. Transverse stability of plane waves is
considered within the framework of this equation. It is shown that the
dispersion terms drastically alter the stability. A necessary stability
condition is obtained in the form of a transverse Benjamin-Feir criterion. If
this condition is met, a quarter of the plane-wave existence band (in terms of
the squared wave number) is unstable, while three quarters are transversely
stable. Next, linear defects in the form of grain boundaries (GB's) are
studied. An effective Burgers equation is derived from the dispersive NWS
equation, in the framework of which a GB is tantamount to a shock wave. It is
shown that the GB's are generic solutions. Asymmetric GB's are moving at a
constant velocity, which is found. The integrability of the Burgers equation
allows one as well to analyze transient processes and interactions between
parallel GB's. The shock-wave solutions obtained in this work may also find
applications in nonlinear fiber optics.
