Integrability of restricted multiple three-wave interactions. II. Coupling constants with ratios 1 and 2

Abstract

The system of ordinary differential equations describing the multiple three-wave interactions with a common pump (daughter) wave is proved to be completely integrable by obtaining the necessary 2N+1 independent first integrals in involution for the case when the coupling constants and the frequency mismatches have ratio 1 and/or 2. This case was deemed integrable on the basis of a Painlevé analysis, but a direct proof has been lacking for some time. The first integrals are the N+1 quadratic Manley–Rowe relations, the cubic Hamiltonian, N−1 quartic integrals (analogous to the ones needed for complete integrability in the case of equal coupling constants and detunings in all wave triads), and a new sixth-order integral involving all wave quantities. The form of this last invariant was deduced from the recent result for the analogous interaction between five waves (N=2), and essentially made possible by the proper use of irreducible forms, elementary building blocks for polynomial first integrals in involution with the Manley–Rowe invariants.