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Solitons and breathers are nonlinear modes that exist in a wide range of physical systems. They are fundamental solutions of a number of nonlinear wave evolution equations, including the uni-directional nonlinear Schrodinger equation (NLSE). We report the observation of slanted solitons and breathers propagating at an angle with respect to the direction of propagation of the wave field. As the coherence is diagonal, the scale in the crest direction becomes finite, consequently, a beam dynamics forms. Spatio-temporal measurements of the water surface elevation are obtained by stereo-reconstructing the positions of the floating markers placed on a regular lattice and recorded with two synchronized high-speed cameras. Experimental results, based on the predictions obtained from the (2D+1) hyperbolic NLSE equation, are in excellent agreement with the theory. Our study proves the existence of such unique and coherent wave packets and has serious implications for practical applications in optical sciences and physical oceanography. Moreover, unstable wave fields in this geometry may explain the formation of directional large amplitude rogue waves with a finite crest length within a wide range of nonlinear dispersive media, such as Bose-Einstein condensates, plasma, hydrodynamics and optics.
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The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to nonconvex flux is studied in the framework of the modified Korteweg-de Vries (mKdV) equation, a canonical model for nonlinear internal gravity wave propagation in stratified fluids. The principal feature of the studied interaction is that both the solitary wave and the large-scale mean flow -- a rarefaction wave or a dispersive shock wave (undular bore) -- are described by the same dispersive hydrodynamic equation. A recent theoretical and experimental study of this new type of dynamic soliton-mean flow interaction has revealed two main scenarios when the solitary wave either tunnels through the varying mean flow that connects two constant asymptotic states, or remains trapped inside it. While the previous work considered convex systems, in this paper it is demonstrated that the presence of a nonconvex hydrodynamic flux introduces significant modifications to the scenarios for transmission and trapping. A reduced set of Whitham modulation equations, termed the solitonic modulation system, is used to formulate a general, approximate mathematical framework for solitary wave-mean flow interaction with nonconvex flux. Solitary wave trapping is conveniently stated in terms of crossing characteristics for the solitonic system. Numerical simulations of the mKdV equation agree with the predictions of modulation theory. The developed theory draws upon general properties of dispersive hydrodynamic partial differential equations, not on the complete integrability of the mKdV equation. As such, the mathematical framework developed here enables application to other fluid dynamic contexts subject to nonconvex flux.
We study the evolution of nonlinear surface gravity water-wave packets developing from modulational instability over an uneven bottom. A nonlinear Schrodinger equation (NLSE) with coefficients varying in space along propagation is used as a reference model. Based on a low-dimensional approximation obtained by considering only three complex harmonic modes, we discuss how to stabilize a one-dimensional pattern in the form of train of large peaks sitting on a background and propagating over a significant distance. Our approach is based on a gradual depth variation, while its conceptual framework is the theory of autoresonance in nonlinear systems and leads to a quasi-frozen state. Three main stages are identified: amplification from small sideband amplitudes, separatrix crossing, and adiabatic conversion to orbits oscillating around an elliptic fixed point. Analytical estimates on the three stages are obtained from the low-dimensional approximation and validated by NLSE simulations. Our result will contribute to understand dynamical stabilization of nonlinear wave packets and the persistence of large undulatory events in hydrodynamics and other nonlinear dispersive media.
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.
We investigate numerically the interactions of two in-phase and out-of-phase Airy beams and nonlinear accelerating beams in Kerr and saturable nonlinear media, in one transverse dimension. We find that bound and unbound soliton pairs, as well as single solitons, can form in such interactions. If the interval between two incident beams is large relative to the width of their first lobes, the generated soliton pairs just propagate individually and do not interact. However, if the interval is comparable to the widths of the maximum lobes, the pairs interact and display varied behavior. In the in-phase case, they attract each other and exhibit stable bound, oscillating, and unbound states, after shedding some radiation initially. In the out-of-phase case, they repel each other and after an initial interaction, fly away as individual solitons. While the incident beams display acceleration, the solitons or soliton pairs generated from those beams do not.
The nonlinear coherent interaction of light with the dispersive and Kerr-type third-order susceptibility medium containing optical impurity atoms or semiconductor quantum dots is considered. Using the generalized perturbation reduction method, the nonlinear wave equation is reduced to the coupled nonlinear Schrodinger equations. It is shown that the second-order derivatives play a key role in the description of the process of formation of the bound state of two breathers oscillating with the sum and the difference of frequencies and wave numbers. The resonant, nonresonant and hybrid mechanisms of the formation of the two-component nonlinear pulse -- the vector breather are realized depending on the light and medium parameters. Explicit analytical expressions for the profile and parameters of the nonlinear pulse are presented. The conditions of the excitation of resonant, nonresonant and hybrid nonlinear waves are discussed. In the particular case, the resonant vector breather coincides with the vector $0pi$ pulse of self-induced transparency.
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