Integrable equations and classical S-matrix

We study amplitudes of five-wave interactions for evolution Hamiltonian equations differ from the KdV equation by the form of dispersion law. We find that five-wave amplitude is canceled for all three known equations (KdV, Benjamin-Ono and equation of intermediate waves) and for two new equations which are natural generalizations of mentioned above.

💡 Research Summary

The paper investigates the integrability of a broad family of one‑dimensional Hamiltonian evolution equations by examining the five‑wave interaction amplitude, a higher‑order nonlinear scattering process that serves as a stringent test for the existence of an infinite hierarchy of conserved quantities. The authors begin by recalling that the classic Korteweg‑de Vries (KdV) equation, with its cubic nonlinearity and third‑order dispersion, is a paradigmatic completely integrable system: it possesses an infinite set of commuting integrals, a Lax pair, and soliton solutions that interact elastically. However, when the dispersion relation is altered—such as in the Benjamin‑Ono equation (dispersion given by the Hilbert transform) or in the so‑called intermediate‑wave equation (a mixture of cubic and fractional‑order dispersion)—the standard inverse‑scattering machinery becomes less transparent, and the question of integrability remains open.

To address this, the authors adopt a classical S‑matrix viewpoint. In a weakly nonlinear, long‑wave regime, the field can be expanded in Fourier modes (u(x,t)=\sum_k a_k(t) e^{ikx}). The linear part supplies a dispersion law (\omega=\Omega(k)); the nonlinear part generates multi‑wave coupling terms. Energy‑ and momentum‑conserving resonances satisfy \