Phase Convergence and Crest Enhancement of Modulated Wave Trains

Abstract

The Akhmediev breather (AB) solution of the nonlinear Schrödinger equation (NLSE) shows that the maximum crest height of modulated wave trains reaches triple the initial amplitude as a consequence of nonlinear long-term evolution. Several fully nonlinear numerical studies have indicated that the amplification can exceed 3, but its physical mechanism has not been clarified. This study shows that spectral broadening, bound-wave production, and phase convergence are essential to crest enhancement beyond the AB solution. The free-wave spectrum of modulated wave trains broadens owing to nonlinear quasi-resonant interaction. This enhances bound-wave production at high wavenumbers. The phases of all the wave components nearly coincide at peak modulation and enhance amplification. This study found that the phase convergence observed in linear-focusing waves can also occur due to nonlinear wave evolution. These findings are obtained by numerically investigating the modulated wave trains using the higher-order spectral method (HOSM) up to the fifth order, which allows investigations of nonlinearity and spectral bandwidth beyond the NLSE framework. Moreover, the crest enhancement is confirmed through a tank experiment wherein waves are generated in the transition region from non-breaking to breaking. Owing to strong nonlinearity, the maximum crest height observed in the tank begins to exceed the HOSM prediction at an initial wave steepness of 0.10.

The nonlinear Benjamin–Feir instability – Hamiltonian dynamics, discrete breathers and steady solutions

“…Such phase coherence has also been observed in simulations, e.g. by Houtani, Sawada & Waseda (2022) and Liu, Waseda & Zhang (2021).…”

“…It is expected that if initial conditions are taken close to , i.e. if some small amount of energy is initially put in the sidebands, then a similar behaviour will be observed in the evolution of the combined phase, as has been recently reported by Houtani et al (2022).…”

Modulational Instability of Nonlinear Wave Packets within (2+4) Korteweg–de Vries Equation

“…Many studies have been devoted to the modulation instability of surface water waves to explain the rogue-wave formation [45][46][47][48][49][50], but in the present study we discuss waves in the interior of the fluid, where the nonlinear instabilities were also predicted [51][52][53][54]. Equation (1) was derived, in particular, for long lowest-mode interfacial waves in a three-layer fluid with an almost symmetric configuration [13,14] in which the lower-order nonlinear terms vanish and higher-order contributions govern the behavior of wave phenomena in the system.…”

“…where l = h/H. Expanding the above Expressions (48) for the coefficients of Equation ( 1) into a Taylor series near the point h cr (δ = (h − h cr )/H) yields As follows from Expressions ( 48)-( 51), the coefficient α2 at the cubic nonlinearity is sign-variable in the vicinity of hcr/H, the coefficient α4 at the quintic nonlinearity is negative in this region, and the dispersion coefficient β is always positive for internal waves. Let us write down the criterion for modulation instability of the wave field and its other characteristics for this physical context.…”

“…In case when both signs are pluses, q > 0 for any values of wave amplitude a0, and modulation instability cannot occur in this zone. Finally, the combination of the signs "+ −" gives q < 0 when Substituting Expressions (48) for the coefficients into expression for q, (52) we obtain:…”