Modulation instability during the propagation of three or four waveguide modes in a thin left-handed film on a right-handed nonlinear substrate

We consider the modulation instability of TE₂ modes propagating in a thin left-handed film on a right-handed Kerr substrate at a frequency near their zero group velocity for the cases of propagation of three and four modes. The dependence of the instability growth rate on the transverse wave number of amplitude mode perturbations is analysed. The features of the modulation instability resulting from the propagation of modes with negative group velocities are determined. The growth rates and regions of instability are compared for various variants of propagation of one, two, three, and four modes.