The Newell-Whitehead-Segel Equation for Traveling Waves
  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. 
Boris Malomed