Since the 1960s, the Benjamin-Feir (or modulation) instability (MI) has been considered as the self-modulation of the continuous “envelope waves” with respect to small periodic perturbations that ...precedes the emergence of highly localized wave structures. Nowadays, the universal nature of MI is established through numerous observations in physics. However, even now, 50 years later, more practical but complex forms of this old physical phenomenon at the frontier of nonlinear wave theory have still not been revealed (i.e., when perturbations beyond simple harmonic are involved). Here, we report the evidence of the broadest class of creation and annihilation dynamics of MI, also called superregular breathers. Observations are done in two different branches of wave physics, namely, in optics and hydrodynamics. Based on the common framework of the nonlinear Schrödinger equation, this multidisciplinary approach proves universality and reversibility of nonlinear wave formations from localized perturbations for drastically different spatial and temporal scales.
We study the nonlinear stage of the modulation instability of a condensate in the framework of the focusing nonlinear Schrödinger equation (NLSE). We find a general N-solitonic solution of the ...focusing NLSE in the presence of a condensate by using the dressing method. We separate a special designated class of "regular solitonic solutions" that do not disturb phases of the condensate at infinity by coordinate. All regular solitonic solutions can be treated as localized perturbations of the condensate. We find an important class of "superregular solitonic solutions" which are small perturbations at a certain moment of time. They describe the nonlinear stage of the modulation instability of the condensate.
We discuss the early history of an important field of “sturm and drang” in modern theory of nonlinear waves. It is demonstrated how scientific demand resulted in independent and almost simultaneous ...publications by many different authors on modulation instability, a phenomenon resulting in a variety of nonlinear processes such as envelope solitons, envelope shocks, freak waves, etc. Examples from water wave hydrodynamics, electrodynamics, nonlinear optics, and convection theory are given.
Lump Interactions with Plane Solitons Stepanyants, Yu. A.; Zakharov, D. V.; Zakharov, V. E.
Radiophysics and quantum electronics,
03/2022, Letnik:
64, Številka:
10
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We analyze in detail the interactions of two-dimensional solitary waves called lumps and onedimensional line solitons within the framework of the Kadomtsev–Petviashvili equation describing wave ...processes in media with positive dispersion. We show that line solitons can emit or absorb lumps or periodic chains of lumps, as well as interact with each other by means of lumps. Within a certain time interval, lumps or lump chains can emerge between two line solitons and then disappear due to absorption by one of the solitons. This phenomenon resembles the appearance of rogue waves in the oceans. The results obtained are graphically illustrated and can be applicable to the description of the physical processes occurring in plasmas, fluids, solids, nonlinear optical media, etc.
We describe a general N-solitonic solution of the focusing nonlinear Schrödinger equation in the presence of a condensate by using the dressing method. We give the explicit form of one- and ...two-solitonic solutions and study them in detail as well as solitonic atoms and degenerate solutions. We distinguish a special class of solutions that we call regular solitonic solutions. Regular solitonic solutions do not disturb phases of the condensate at infinity by coordinate. All of them can be treated as localized perturbations of the condensate. We find a broad class of superregular solitonic solutions which are small perturbations at a certain moment of time. Superregular solitonic solutions are generated by pairs of poles located on opposite sides of the cut. They describe the nonlinear stage of the modulation instability of the condensate and play an important role in the theory of freak waves.
In the framework of the focusing nonlinear Schrödinger equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. The development of the MI leads to the ...formation of 'integrable turbulence' (Zakharov 2009 Stud. Appl. Math. 122 219-34). We study the time evolution of its major characteristics averaged across realizations of initial data-the condensate solution seeded by small random noise with fixed statistical properties. We observe that the system asymptotically approaches to the stationary integrable turbulence, however this is a long process. During this process momenta, as well as kinetic and potential energies, oscillate around their asymptotic values. The amplitudes of these oscillations decay with time t as t−3/2, the phases contain the nonlinear phase shift that decays as t−1/2, and the frequency of the oscillations is equal to the double maximum growth rate of the MI. The evolution of wave-action spectrum is also oscillatory, and characterized by formation of the power-law region ∼|k|−α in the small vicinity of the zeroth harmonic k = 0 with exponent α close to 2/3. The corresponding modes form 'quasi-condensate', that acquires very significant wave action and macroscopic potential energy. The probability density function of wave amplitudes asymptotically approaches the Rayleigh distribution in an oscillatory way. Nevertheless, in the beginning of the nonlinear stage the MI slightly increases the occurrence of rogue waves. This takes place at the moments of potential energy modulus minima, where the PDF acquires 'fat tales' and the probability of rogue waves occurrence is by about two times larger than in the asymptotic stationary state. Presented facts need a theoretical explanation.
An interface between a dielectric and a medium containing charge carriers with moderate mobility is considered. In equilibrium, stochastic fluxes of positive and negative particles toward the surface ...have equal average current density j 0, and we suppose that the surface absorbs all falling charges. All over the surface, this results in the emergence of oppositely charged spots of various sizes D fluctuating and interacting with each other. Fourier expansion reduces this collection of interacting spots to the ensemble of independently fluctuating charge density waves. An exact solution of the Poisson equation for a single wave on a flat surface was obtained and provided strict proof that a fluctuating electric field is quite strong just above each charge spot but diminishes exponentially with the distance from the plane. The lifetime τ of a charge spot is inversely proportional to the density j 0 of the stochastic current while proportional to j 0 τ fluctuation’s amplitudes independent of j 0. The fluctuation’s parameter dependence on the charge spot’s size D can vary according to the conducting medium properties.
The Late Jurassic – earliest Cretaceous time interval was characterized by a widespread distribution of dysoxiс–anoxiс environments in temperate- and high-latitude epicontinental seas, which could be ...defined as a shelf dysoxic–anoxic event (SDAE). In contrast to black shales related to oceanic anoxic events, deposits generated by the SDAE were especially common in shelf sites in the Northern Hemisphere. The onset and termination of the SDAE was strongly diachronous across different regions. The SDAE was not associated with significant disturbances of the carbon cycle. Deposition of organic-carbon-rich sediment and the existence of dysoxic–anoxic conditions during the SDAE lasted up to c. 20 Ma, but this event did not cause any remarkable biotic extinction. Temperate- and high-latitude black shale occurrences across the Jurassic–Cretaceous boundary have been reviewed. Two patterns of black shale deposition during the SDAE are recognized: (1) Subboreal type, with numerous thin black shale beds, bounded by sediments with very low total organic carbon (TOC) values; and (2) Boreal type, distinguished by predominantly thick black shale successions showing high TOC values and prolonged anoxic–dysoxic conditions. These types appear to be unrelated to differences in accommodation space, and can be clearly recognized irrespective of the thickness of shale-bearing units. Black shales in high-latitude areas in the Southern Hemisphere strongly resemble Boreal types of black shale by their mode of occurrence. The causes of this SDAE are linked to long-term warming and changes in oceanic circulation. Additionally, the long-term disturbance of planktonic communities may have triggered overall increased productivity in anoxia-prone environments.
Super compact equation for water waves Dyachenko, A. I.; Kachulin, D. I.; Zakharov, V. E.
Journal of fluid mechanics,
10/2017, Letnik:
828
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Mathematicians and physicists have long been interested in the subject of water waves. The problems formulated in this subject can be considered fundamental, but many questions remain unanswered. For ...instance, a satisfactory analytic theory of such a common and important phenomenon as wave breaking has yet to be developed. Our knowledge of the formation of rogue waves is also fairly poor despite the many efforts devoted to this subject. One of the most important tasks of the theory of water waves is the construction of simplified mathematical models that are applicable to the description of these complex events under the assumption of weak nonlinearity. The Zakharov equation, as well as the nonlinear Schrödinger equation (NLSE) and the Dysthe equation (which are actually its simplifications), are among them. In this article, we derive a new modification of the Zakharov equation based on the assumption of unidirectionality (the assumption that all waves propagate in the same direction). To derive the new equation, we use the Hamiltonian form of the Euler equation for an ideal fluid and perform a very specific canonical transformation. This transformation is possible due to the ‘miraculous’ cancellation of the non-trivial four-wave resonant interaction in the one-dimensional wave field. The obtained equation is remarkably simple. We call the equation the ‘super compact water wave equation’. This equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. The NLSE and the Dysthe equations (Dysthe Proc. R. Soc. Lond. A, vol. 369, 1979, pp. 105–114) can be easily derived from the super compact equation. This equation is also suitable for analytical studies as well as for numerical simulation. Moreover, this equation also allows one to derive a spatial version of the water wave equation that describes experiments in flumes and canals.
We study numerically the nonlinear stage of the modulational instability (MI) of cnoidal waves in the framework of the focusing one-dimensional nonlinear Schrödinger (NLS) equation. Cnoidal waves are ...exact periodic solutions of the NLS equation which can be represented as the lattices of overlapping solitons. The MI of these lattices leads to the development of 'integrable turbulence' (Zakharov 2009 Stud. Appl. Math. 122 219-34). We study the major characteristics of turbulence for the dn-branch of cnoidal waves and demonstrate how these characteristics depend on the degree of 'overlapping' between the solitons within the cnoidal wave. Integrable turbulence, which develops from the MI of the dn-branch of cnoidal waves, asymptotically approaches its stationary state in an oscillatory way. During this process, kinetic and potential energies oscillate around their asymptotic values. The amplitudes of these oscillations decay with time as t−α, 1<α<1.5, the phases contain nonlinear phase shift decaying as t−1/2, and the frequency of the oscillations is equal to the double maximal growth rate of the MI, s=2γmax. In the asymptotic stationary state, the ratio of potential to kinetic energy is equal to −2. The asymptotic PDF of the wave intensity is close to the exponential distribution for cnoidal waves with strong overlapping, and is significantly non-exponential for cnoidal waves with weak overlapping of the solitons. In the latter case, the dynamics of the system reduces to two-soliton collisions, which occur at an exponentially small rate and provide an up to two-fold increase in amplitude compared with the original cnoidal wave. For all cnoidal waves of the dn-branch, the rogue waves at the time of their maximal elevation have a quasi-rational profile similar to that of the Peregrine solution.
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