Reductive Perturbation Method for Wave-Modulation in Multi-Dimensional Space

The reductive perturbation method for weakly nonlinear waves propagating in strongly dispersive media is applied to systems of more than one space variable. A nonlinear Klein-Gordon equation with two space variables ( x and z ) is considered as a model equation, and two cases are discussed. In the first case, the system is bounded in the z -derection so that waves propagate through a wave-guide along the x -axis, in which the unperturbed state is a trivial constant state. In the next case, the unperturbed state is not constant but steeply decreases for | z |→∞. In both cases wave-modulations in the x -direction can be described by a non-linear Schrödinger equation. A general system of interacting Bose fields is also considered and slow temporal oscillations of a radially decreasing (particle-like) solution is examined.