Transverse Instability for Periodic Waves of Kp-i and Schrödinger Equations

We consider the quadratic and cubic KP I and NLS models in 1+2 dimensions with periodic boundary conditions. We show that the spatially periodic travelling waves (with period K) in the form u(t, x, y) = φ(x− ct) are spectrally and linearly unstable, when the perturbations are taken to be with the same period. This strong instability implies other instabilities considered recently for example with respect to perturbations with periods nK, n = 2, 3, . . . or bounded perturbations.