Envelope soliton resonances and Broer–Kaup-type non-Madelung fluids

We derive an extended nonlinear dispersion for envelope soliton equations and also find generalized equations of the nonlinear Schrödinger (NLS) type associated with this dispersion. We show that space dilatations imply hyperbolic rotation of the pair of dual equations, the NLS and resonant NLS (RNLS) equations. For the RNLS equation, in addition to the Madelung fluid representation, we find an alternative non-Madelung fluid system in the form of a Broer–Kaup system. Using the bilinear form for the RNLS equation, we construct the soliton resonances for the Broer–Kaup system and find the corresponding integrals of motion and existence conditions for the soliton resonance and also a geometric interpretation in terms of a pseudo-Riemannian surface of constant curvature. This approach can be extended to construct a resonance version and the corresponding Broer–Kaup-type representation for any envelope soliton equation. As an example, we derive a new modified Broer–Kaup system from the modified NLS equation.