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We present experimental observations of the hierarchy of rational breather solutions of the nonlinear Schrödinger equation (NLS) generated in a water wave tank. First, five breathers of the infinite hierarchy have been successfully generated, thus confirming the theoretical predictions of their existence. Breathers of orders higher than five appeared to be unstable relative to the wave-breaking effect of water waves. Due to the strong influence of the wave breaking and relatively small carrier steepness values of the experiment these results for the higher-order solutions do not directly explain the formation of giant oceanic rogue waves. However, our results are important in understanding the dynamics of rogue water waves and may initiate similar experiments in other nonlinear dispersive media such as fiber optics and plasma physics, where the wave propagation is governed by the NLS.

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The role of multiple soliton and breather interactions in formation of very high waves is disclosed within the framework of integrable modified Korteweg -de Vries (mKdV) equation. Optimal conditions for the focusing of many solitons are formulated explicitly. Namely, trains of ordered solitons with alternate polarities evolve to huge strongly localized transient waves. The focused wave amplitude is exactly the sum of the focusing soliton heights; the maximum wave inherits the polarity of the fastest soliton in the train. The focusing of several solitary waves or/and breathers may naturally occur in a soliton gas and will lead to rogue-wave-type dynamics; hence it represents a new nonlinear mechanism of rogue wave generation. The discovered scenario depends crucially on the soliton polarities (phases), and thus cannot be taken into account by existing kinetic theories. The performance of the soliton mechanism of rogue wave generation is shown for the example of focusing mKdV equation, when solitons possess 'frozen' phases (polarities), though the approach works in other integrable systems which admit soliton and breather solutions. Introduction.-The generation of unexpectedly large waves from stochastic fields has attracted much interest in many recent studies thanks to recognition of the rogue wave phenomenon in marine and optical realms (see [1] and many others). The modulational instability is the most recognized physical effect capable of generation of very high waves due to the energy transfer from many waves towards an inoculating perturbation of the wavetrain. Most importantly, the nonlinear dynamics alters essentially the statistical properties of stochastic waves, favouring occurrence of very high waves. The accounting for significant deviations from the quasi Gaussian states breaks down the classic assumptions of the wave turbulence theory. The wave phase averaging becomes inappropriate, thus direct simulations of irregular waves are involved to discover the statistics of high waves.