The Transverse Instability of Periodic Waves in Zakharov–Kuznetsov Type Equations

Abstract

In this paper, we investigate the instability of one‐dimensionally stable periodic traveling wave solutions of the generalized Korteweg‐de Vries equation to long wavelength transverse perturbations in the generalized Zakharov–Kuznetsov equation in two space dimensions. By deriving appropriate asymptotic expansions of the periodic Evans function, we derive an index which yields sufficient conditions for transverse instabilities to occur. This index is geometric in nature, and applies to any periodic traveling wave profile under some minor smoothness assumptions on the nonlinearity. We also describe the analogous theory for periodic traveling waves of the generalized Benjamin–Bona–Mahony equation to long wavelength transverse perturbations in the gBBM–Zakharov–Kuznetsov equation.