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We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrodinger (NLS) equation with the initial condition in the form of a rectangular barrier (a box). We use the Whitham modulation theory combined with the nonlinear steepest descent for the semi-classical inverse scattering transform, to describe the evolution and interaction of two counter-propagating nonlinear wave trains --- the dispersive dam break flows --- generated in the NLS box problem. We show that the interaction dynamics results in the emergence of modulated large-amplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain space-time domain. Our semi-classical analytical results are shown to be in excellent agreement with the results of direct numerical simulations of the small-dispersion focusing NLS equation.
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Solitons and breathers are localized solutions of integrable systems that can be viewed as particles of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media these integrable gases present fundamental interest for nonlinear physics. We develop analytical theory of breather and soliton gases by considering a special, thermodynamic type limit of the wavenumber-frequency relations for multi-phase (finite-gap) solutions of the focusing nonlinear Schrodinger equation. This limit is defined by the locus and the critical scaling of the band spectrum of the associated Zakharov-Shabat operator and yields the nonlinear dispersion relations for a spatially homogeneous breather or soliton gas, depending on the presence or absence of the background Stokes mode. The key quantity of interest is the density of states defining, in principle, all spectral and statistical properties of a soliton (breather) gas. The balance of terms in the nonlinear dispersion relations determines the nature of the gas: from an ideal gas of well separated, non-interacting breathers (solitons) to a special limiting state, which we term breather (soliton) condensate, and whose properties are entirely determined by the pairwise interactions between breathes (solitons). For a non-homogeneous breather gas, we derive a full set of kinetic equations describing slow evolution of the density of states and of its carrier wave counterpart. The kinetic equation for soliton gas is recovered by collapsing the Stokes spectral band. A number of concrete examples of breather and soliton gases are considered, demonstrating efficacy of the developed general theory with broad implications for nonlinear optics, superfluids and oceanography.
Rogue waves appearing on deep water or in optical fibres are often modelled by certain breather solutions of the focusing nonlinear Schrodinger (fNLS) equation which are referred to as solitons on finite background (SFBs). A more general modelling of rogue waves can be achieved via the consideration of multiphase, or finite-band, fNLS solutions of whom the standard SFBs and the structures forming due to their collisions represent particular, degenerate, cases. A generalised rogue wave notion then naturally enters as a large-amplitude localised coherent structure occurring within a finite-band fNLS solution. In this paper, we use the winding of real tori to show the mechanism of the appearance of such generalized rogue waves and derive an analytical criterion distinguishing finite-band potentials of the fNLS equation that exhibit generalised rogue waves.
The double-periodic solutions of the focusing nonlinear Schrodinger equation have been previously obtained by the method of separation of variables. We construct these solutions by using an algebraic method with two eigenvalues. Furthermore, we characterize the Lax spectrum for the double-periodic solutions and analyze rogue waves arising on their background. Magnification of the rogue waves is studied numerically.
We show that the nonlinear stage of modulational instability induced by parametric driving in the {em defocusing} nonlinear Schrodinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearised Floquet analysis.
We numerically realize breather gas for the focusing nonlinear Schrodinger equation. This is done by building a random ensemble of N $sim$ 50 breathers via the Darboux transform recursive scheme in high precision arithmetics. Three types of breather gases are synthesized according to the three prototypical spectral configurations corresponding the Akhmediev, Kuznetsov-Ma and Peregrine breathers as elementary quasi-particles of the respective gases. The interaction properties of the constructed breather gases are investigated by propagating through them a trial generic breather (Tajiri-Watanabe) and comparing the mean propagation velocity with the predictions of the recently developed spectral kinetic theory (El and Tovbis, PRE 2020).
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