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The Akhmediev breather (AB) solution of the nonlinear Schr$\"{o}$dingerequation (NLSE) shows that the maximum crest height of modulated wave trainsreaches triple the initial amplitude as a consequence of nonlinear long-termevolution. Several fully nonlinear numerical studies have indicated that theamplification can exceed 3, but its physical mechanism has not been clarified.This study shows that spectral broadening, bound-wave production, and phaseconvergence are essential to crest enhancement beyond the AB solution. Thefree-wave spectrum of modulated wave trains broadens owing to nonlinearquasi-resonant interaction. This enhances bound-wave production at highwavenumbers. The phases of all the wave components nearly coincide at peakmodulation and enhance amplification. We find that the phase convergenceobserved in linear-focusing waves can also occur due to nonlinear waveevolution. These findings are obtained by numerically investigating themodulated wave trains using the higher-order spectral method (HOSM) up to thefifth order, which allows investigations of nonlinearity and spectral bandwidthbeyond the NLSE framework. Moreover, we confirm the crest enhancement through atank experiment wherein waves are generated in the transition region fromnon-breaking to breaking. Owing to strong nonlinearity, the maximum crestheight observed in the tank begins to exceed the HOSM prediction at an initialwave steepness of 0.10.

Phase convergence and crest enhancement of modulated wave trains