Free-Surface Hydrodynamics in Conformal Variables: Are Equations of Free-Surface Hydrodynamics on Deep Water Integrable?

The hypothesis on complete integrability of equations describing the potential motion of incompressible ideal fluid with free surface in 2-D space in presence and absence of gravity was formulated by Dyachenko and Zakharov in 1994 [1]. Later on, the same authors found that these equations have indefinite number of additional motion constants [2] that was an argument in support of the integrability hypothesis. In this article we formulate another argument in favor of this conjecture. It is known [3] that the free-surface equations have an exotic solution that keeps the surface flat but describes the compression of the whole mass of fluid. In this article we show that the free-surface hydrodynamic is integrable if the motion can be treated as a finite amplitude perturbation of the compressed fluid solution. Integrability makes possible to construct an indefinite number of exact solutions of the Euler equations with free surface.Comment: 8 pages, 7 references. The content of this article was reported on VIIIth International Lavrentyev Readings, September 6-11, 2015, Novosibirsk, Russi

Direct and Inverse Cascades in the Wind-Driven Sea

We offer a new form for the S(nl) term in the Hasselmann kinetic equation for squared wave amplitudes of wind-driven gravity wave. This form of S(nl) makes possible to rewrite in differential form the conservation laws for energy, momentum, and wave action, and introduce their fluxes by a natural way. We show that the stationary kinetic equation has a family of exact Kolmogorov-type solutions governed by the fluxes of motion constants: wave action, energy, and momentum. The simple "local" model for S(nl) term that is equivalent to the "diffusion approximation" is studied in details. In this case, Kolmogorov spectra are found in the explicit form. We show that a general solution of the stationary kinetic equation behind the spectral peak is described by the Kolmogorov-type solution with frequency-dependent fluxes. The domains of "inverse cascade" and "direct cascade" can be separated by natural way. The spectrum in the universal domain is close to $\omega^{-4}$

Non-periodic one-gap potentials in quantum mechanics

We describe a broad class of bounded non-periodic potentials in one-dimensional stationary quantum mechanics having the same spectral properties as periodic potentials. The spectrum of the corresponding Schroedinger operator consists of a finite or infinite number of allowed bands separated by gaps. In this letter we consider the simplest class of potentials, whose spectra consist of an interval on the negative semiaxis and the entire positive axis. The potentials are reflectionless, and a particle with positive energy moves freely in both directions. The potential is constructed as a limit of Bargmann potentials and is determined by a Riemann-Hilbert problem, which is equivalent to a pair of singular integral equations that can be efficiently solved using numerical techniques

Rogue waves statistics in the framework of one-dimensional Generalized Nonlinear Schrodinger Equation

We measure evolution of spectra, spatial correlation functions and probability density functions (PDFs) of waves appearance for a set of one-dimensional NLS-like equations of focusing type, namely for the classical integrable Nonlinear Schrodinger equation (1), nonintegrable NLS equation accounting for dumping (linear dissipation, two- and three-photon absorption) and pumping terms (2) and generalized NLS equation accounting for six-wave interactions, dumping and pumping terms (3). All additional terms beyond the classical NLS equation are small. As initial conditions we choose seeded by noise modulationally unstable solutions of the considered systems in the form of (a) condensate for systems (1)-(3) and (b) cnoidal wave for the classical NLS equation (1). We observe 'strange' results for the classical NLS equation (1) with condensate initial condition including peak at zeroth harmonic in averaged over ensemble spectra, non-decaying spacial correlation functions and a 'breathing' region on the PDFs for medium waves amplitudes where frequency of waves appearance oscillates with time, while the far-tails of the PDFs remain Rayleigh ones. Addition of small dumping and pumping terms in model (2) breaks integrability that results in absence of the peak at zeroth harmonic in spectra, spacial correlation functions decaying to zero level and strictly Rayleigh PDFs for waves amplitudes. For the classical NLS equation (1) with cnoidal wave initial condition PDFs turn out to be significantly different from Rayleigh ones with 'fat tails' in the region of large amplitudes where higher waves appear more frequently, while for generalized NLS equation with six-wave interactions, dumping and pumping terms (3) we demonstrate absence of non-Rayleigh addition to the PDFs for zeroth six-wave interactions coefficient and increase of non-Rayleigh addition with six-wave interactions term.Comment: 20 pages, 47 figure

Limited fetch revisited: comparison of wind input terms, in surface waves modeling

Results pertaining to numerical solutions of the Hasselmann kinetic equation (HE), for wind driven sea spectra, in the fetch limited geometry, are presented. Five versions of source functions, including the recently introduced ZRP model, have been studied, for the exact expression of Snl and high-frequency implicit dissipation, due to wave-breaking. Four of the five experiments were done in the absence of spectral peak dissipation for various Sin terms. They demonstrated the dominance of quadruplet wave-wave interaction, in the energy balance, and the formation of self-similar regimes, of unlimited wave energy growth, along the fetch. Between them was the ZRP model, which strongly agreed with dozens of field observations performed in the seas and lakes, since 1947. The fifth, the WAM3 wind input term experiment, used additional spectral peak dissipation and reproduced the results of a previous, similar, numerical simulation, but only supported the field experiments for moderate fetches, demonstrating a total energy saturation at half of that of the Pierson-Moscowits limit. The alternative framework for HE numerical simulation is proposed, along with a set of tests, allowing one to select physically-justified source terms

On nonlinearity implications and wind forcing in Hasselmann equation

We discuss several experimental and theoretical techniques historically used for Hasselmann equation wind input terms derivation. We show that recently developed ZRP technique in conjunction with high-frequency damping without spectral peak dissipation allows to reproduce more than a dozen of fetch-limited field experiments. Numerical simulation of the same Cauchy problem for different wind input terms has been performed to discuss nonlinearity implications as well as correspondence to theoretical predictions

Soliton on Unstable Condensate

We construct new exact solutions of the focusing Nonlinear Schr\"{o}dinger equation (NLSE). This is a soliton propagating on an unstable condensate. The Kuznetsov and Akhmediev solitons as well as the Peregrine instanton are particular cases of this new solution. We discuss applications of this new solution to the description of freak (rogue) waves in the ocean and in optical fibers.Comment: 4 pages, 3 figure

Transparency of Strong Gravitational Waves

This paper studies diagonal spacetime metrics. It is shown that the overdetermined Einstein vacuum equations are compatible if one Killing vector exists. The stability of plane gravitational waves of the Robinson type is studied. This stability problem bares a fantastic mathematical resemblance to the stability of the Schwarzschild black hole studied by Regge and Wheeler. Just like for the Schwarzschild black hole, the Robinson gravitational waves are proven to be stable with respect to small perturbations. We conjecture that a bigger class of vacuum solutions are stable, among which are all gravitational solitons. Moreover, the stability analysis reveals a surprising fact: a wave barrier will be transparent to the Robinson waves, which therefore passes through the barrier freely. This is a hint of integrability of the 1+2 vacuum Einstein equations for diagonal metrics

Nonlinear Ocean Waves Amplification in Straits

We study deep water ocean wind-driven waves in strait, with wind directed orthogonally to the shore, through exact Hasselmann equation. Despite of "dissipative" shores - we do not include any reflection from the coast lines - we show that the wave turbulence evolution can be split in time into two different regimes. During the first wave propagate along the wind, and the wind-driven sea can be described by the self-similar solution of the Hasselmann equation like in the open sea. The second regime starts later in time, after significant enough wave energy accumulation at the down-wind boundary. Since this moment the ensemble of waves propagating against the wind starts its formation. Also, the waves, propagation along the strait start to appear. The wave system eventually reaches asymptotic stationary state in time, consisting of two co-existing states: the first, self-similar wave ensemble, propagating with the wind, and the second, quasi-monochromatic waves, propagating almost orthogonal to the wind direction and tending to slant against the wind at the angle of 15 degrees closer to the wave turbulence origination shore line. These "secondary waves" appear only due to intensive nonlinear wave interaction. The total wave energy exceeds its "expected value" approximately by the factor of two, with respect to estimated in the absence of the shores. It is expected that in the reflective shores presence this amplification will grow essentially. We propose to call this laser-like Nonlinear Ocean Waves Amplification mechanism by the acronym NOWA

Generalized primitive potentials

In our previous work, we introduced a new class of bounded potentials of the one-dimensional Schr\"odinger operator on the real axis, and a corresponding family of solutions of the KdV hierarchy. These potentials, which we call primitive, are obtained as limits of rapidly decreasing reflectionless potentials, or multisoliton solutions of KdV. In this note, we introduce generalized primitive potentials, which are obtained as limits of all rapidly decreasing potentials of the Schr\"odinger operator. These potentials are constructed by solving a contour problem, and are determined by a pair of positive functions on a finite interval and a functional parameter on the real axis.Comment: 10 page