On the integrable six-wave interaction system and its space-time shifted reduction

Mark J. Ablowitz; Ramesh Gupta; Ziad H. Musslimani; Nicholas J. Ossi

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arXiv · 2510.10312

On the integrable six-wave interaction system and its space-time shifted reduction

Mark J. Ablowitz·Ramesh Gupta·Ziad H. Musslimani·Nicholas J. Ossi

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Abstract

The multi-dimensional six-wave interaction system is derived in the context of nonlinear optics. Starting from Maxwell's equations, a reduced system of equations governing the dynamics of the electric and polarization fields are obtained. Using a space-time multi-scale asymptotic expansion, a hierarchy of coupled equations describing the spatio-temporal evolution of the perturbed electric and polarization fields are derived. The leading order equation admits a six-wave ansatz satisfying a triad resonance condition. By removing secular terms at next order, a first order in space and time quadratically nonlinear coupled six-wave interaction system is obtained. This resulting system is tied to its integrable counterpart which was postulated by Ablowitz and Haberman in the 1970s. A reduction to a space-time shifted nonlocal three-wave system is presented. The resulting system is solved using the inverse scattering transform, which employs nonlocal symmetries between the associated eigenfunctions and scattering data; soliton solutions are then obtained. Finally, an infinite set of conservation laws for the six-wave system is derived; one is shown to be connected to its Hamiltonian structure.

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Mark J. Ablowitz, Ramesh Gupta, Ziad H. Musslimani, Nicholas J. Ossi. 2025-10-11. On the integrable six-wave interaction system and its space-time shifted reduction. https://arxiv.org/abs/2510.10312

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