Stability and grain boundaries in the dispersive Newell-Whitehead-Segel equation

An equation to describe nearly Ip traveling-waves patterns, obtained first by Brand, Lomdahl and Newell, is revisited. In addition to the previously known transverse Benjamin-Feir condition, necessary for stability of plane waves, it is demonstrated that, if this condition is met, a quarter of the plane-wave existence band is unstable against transverse perturbations, white three quarters are stable. Most essential results are obtained for linear defects in the form of grain boundaries (GB's). An effective Burgers equation is derived, in the framework of which a GB is tantamount to a shock wave. Asymmetric GB's are moving at a constant velocity.