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The main observation of this paper it that the modified Korteweg-de Vries equation has its natural origin in phase modulation of a basic state such as a periodic travelling wave or more generally a family of relative equilibria. Extension to 2+1 suggests that a modified Kadomtsev-Petviashvili (or a Konopelchenko-Dubrovsky) equation should emerge, but our result shows that there is an additional term which has gone heretofore unnoticed. Thus through the novel application of phase modulation a new equation appears as the 2+1 extension to a previously known one. To demonstrate the theory it is applied to the cubic-quintic Nonlinear Schrodinger (CQNLS) equation, showing that there are relevant parameter values where a modified KP equation bifurcates from periodic travelling wave solutions of the 2+1 CQNLS equation.

Reduction to modified KdV and its KP-like generalization via phase modulation