Recurrence in the high-order nonlinear Schrödinger equation: A low-dimensional analysis

Abstract

We study a three-wave truncation of the high-order nonlinear Schrödinger equation for deepwater waves (HONLS, also named Dysthe equation). We validate the model by comparing it to numerical simulation, we distinguish the impact of the different fourth-order terms and classify the solutions according to their topology. This allows us to properly define the temporary spectral upshift occurring in the nonlinear stage of Benjamin-Feir instability and provides a tool for studying further generalizations of this model.

Nonlinear stage of Benjamin-Feir instability in forced/damped deep-water waves

“…We apply to (2) the three-wave truncation approach [34, p. 527] and briefly recall, as a reference, its results [25] in the case of no damping/forcing. At the beginning of the wave-tank we excite a homogeneous carrier wave perturbed by a pure oscillation at normalized angular frequency Ω.…”

“…As described in Ref. [25] and Appendix C, in the case of δ 0 = δ 1 = 0, despite the broken integrability of the reduced system (α is not conserved), the perturbed evolution is still quasi-periodic and can be conveniently mapped on the Hamiltonian level sets of the corresponding integrable system. The heteroclinic structure is summarized in Fig.…”

“…We showed in Ref. [25] that the three wave truncation conserves N 3 and α, thus the system is reduced-without loss of generality we let N 3 = 1-to a one degree-of-freedom system in the conjugate variables (ψ, η) with Hamiltonian function (η = ∂H ∂ψ ,ψ = − ∂H ∂η ), with Hamiltonian…”

“…Finally, we should not forget that exact solutions, such as the Akhmediev breather (AB) [32,33], are not available in the HONLS case; we rely instead on a three-mode truncation [25,34,35]. This approach provides good approximated solutions and allows a more flexible choice of initial conditions and range of parameters than conventional approximated ABs [13,15,36].…”

On the incomplete recurrence of modulationally unstable deep-water surface gravity waves

“…However, the recurrence was always approximate in contrast to the exact breather solution of the NLS equation. The effect of high-order terms in the NLS equation on the recurrence process was studied in a recent paper [25] (see also references on the previous research therein).…”