On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation

One-dimensional solitary wave solutions of the Kadomtsev-Petviashvili (KP) equation were shown to be unstable to long-wavelength transverse disturbances by Kadomtsev and Petviashvili, in the positive dispersion case, Here we show that there is a short wavelength cutoff for the instability, which is associated with a bifurcation to transversely modulated solitary waves, and we identify the dominant mode of instability, by finding explicitly all the exponentially unstable modes of the linearized equation for perturbations of the solitary wave. No unstable modes are found in the negative dispersion case.