On the reduction of coupled NLS equations to non-linear phase equations via modulation of a two-phase wavetrain

The phase dynamics of two phase wavetrains in the coupled non-linear Schrödinger (NLS) equations are investigated as an example of the dispersion arising from singular wave action. It is shown that when the wavetrain becomes singular, there is a reduction from coupled NLS to a scalar Korteweg–de Vries (KdV) equation, and if there is a further degeneracy the scalar two-way Boussinesq emerges. This is the first such derivation of the two-way Boussinesq reduction in this setting. A novelty in the theory is that the coefficients in the resulting equations are determined from properties of the wavetrain and underlying conservation laws. This theory generalizes the reduction from a single defocussing NLS equation to the KdV equation, and introduces Boussinesq dynamics to finite amplitude states in this family. A discussion of the effect of the phase dynamics on the wavetrain solution shows that the reductions provide an insight into a mechanism for the bifurcation of periodic wavetrains to dark and bright solitary waves.