Long-living coherent patterns in the fields of irregular waves in deep water

<p>Long-living coherent wave patterns embedded into the irregular wave fields are studied using the data of extensive numerical simulations of the Euler equations in deep water. The distributions of the rogue wave lifetimes according to the numerical simulations of JONSWAP waves with narrow and broad angle spectra are discussed. The observation of a wave group persisting for more than 200 periods in the direct numerical simulation of nonlinear unidirectional irregular water waves is discussed. Through solution of the associated scattering problem for the nonlinear Schrodinger equation, the persisting group is identified as the intense envelope soliton with remarkably stable parameters. Most of extreme waves occur on top of this group, resulting in higher and longer rogue wave events. It is shown that the persisting wave structure survives under the conditions of directional waves with moderate spread of directions. The survivability of coherent wave patterns is expected to further increase when the waves are guided by currents or the topography.</p><p> </p><p>The research is supported by the RSF grant No. 19-12-00253; the study of trapped waves is performed for the RFBR grant No. 21-55-15008.</p>

Modulational Stability of Envelope Soliton in a Quantum Electron-Ion Plasma in Three Dimension - A generalised Nonlinear Schr ̈odinger Equ ation in 3D

Modulational stability of envelope soliton is studied in a quantum dusty plasma in three dimension.<br>The Krylov-Bogoliubov-Mitropolsky method is applied to the three dimension plasma governing<br>equations. A generalised form of Nonlinear Schr¨odinger equation is obtained whose dispersive term<br>has a tensorial character. Stability condition is deduced abintio and the stability zones are plotted<br>as a function of plasma parameters

Modulational Stability of Envelope Soliton in a Quantum Electron-Ion Plasma in Three Dimension - A generalised Nonlinear Schr ̈odinger Equ ation in 3D

Modulational stability of envelope soliton is studied in a quantum dusty plasma in three dimension.<br>The Krylov-Bogoliubov-Mitropolsky method is applied to the three dimension plasma governing<br>equations. A generalised form of Nonlinear Schr¨odinger equation is obtained whose dispersive term<br>has a tensorial character. Stability condition is deduced abintio and the stability zones are plotted<br>as a function of plasma parameters

Breather’s Properties within the Framework of the Modified Korteweg–de Vries Equation

We study a breather’s properties within the framework of the modified Korteweg–de Vries (mKdV) model, where cubic nonlinearity is essential. Extrema, moments, and invariants of a breather with different parameters have been analyzed. The conditions in which a breather moves in one direction or another has been determined. Two limiting cases have been considered: when a breather has an N-wave shape and can be interpreted as two solitons with different polarities, and when a breather contains many oscillations and can be interpreted as an envelope soliton of the nonlinear Schrödinger equation (NLS).

Transformation of envelope solitons propagating over a bottom step

&lt;p&gt;The problem of the weakly nonlinear wave transformation on a bottom step is studied analytically and numerically by means of the direct simulation of the Euler equation. It is assumed that the quasi-linear wave packets can be described by the nonlinear Schr&amp;#246;dinger equation for surface waves in finite-depth water. The process of wave transformation in the vicinity of the bottom step can be described within the framework of the linear theory and the transformation coefficients (the transmission and reflection coefficients) can be determined by the approximate formula suggested in [1]. The fate of transmitted and reflected wave trains emerging from the incident envelope soliton can be determined with the help of the Inverse Scattering Technique [2, 3].&lt;/p&gt;&lt;p&gt;The parameters of secondary envelope solitons (their number, amplitudes, and speeds) asymptotically forming in the far-field zone are obtained analytically and compared against the numerically calculated ones, as the functions of the depth drop &lt;em&gt;h&lt;/em&gt;&lt;sub&gt;2&lt;/sub&gt;/&lt;em&gt;h&lt;/em&gt;&lt;sub&gt;1&lt;/sub&gt;, where &lt;em&gt;h&lt;/em&gt;&lt;sub&gt;1&lt;/sub&gt; and &lt;em&gt;h&lt;/em&gt;&lt;sub&gt;2&lt;/sub&gt;&amp;#160;are the undisturbed water depths in front of and behind the bottom step, respectively. It is shown that the wave amplitudes can notably increase when the envelope soliton travels from the relatively shallow to much deeper water. The amplitudes of secondary solitons can exceed more than twice the amplitude of the incident wave.&lt;/p&gt;&lt;p&gt;The direct numerical simulation of envelope soliton transformation was undertaken by means of the High Order Spectral Method [4, 5]. The comparison of approximate analytical solutions with the results of numerical simulations reveals the domains of very good agreement between the data where the approximate theory is applicable. In the meantime, the noticeable disagreement between the approximate nonlinear theory and the direct simulations is found when the theory is inapplicable.&lt;/p&gt;&lt;p&gt;The research by A.S. is supported by the RFBR grant No. 18-02-00042; he also acknowledges the support from the International Visitor Program of the University of Sydney and is grateful for the hospitality of the University of Southern Queensland. The research of Y.S. was support by the grant of the President of the Russian Federation for State support of scientific research of leading scientific Schools of the Russian Federation NSh-2485.2020.5.&lt;/p&gt;&lt;p&gt;[1] Kurkin, A.A., Semin, S.V., and Stepanyants, Yu.A., Transformation of Surface Waves over a Bottom Step. Izvestiya, Atmospheric and Oceanic Physics, 2015, Vol. 51, 214&amp;#8211;223.&lt;/p&gt;&lt;p&gt;[2] Zakharov, V.E., Shabat, A.B., Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP, 1972, Vol. 34, 62-69.&lt;/p&gt;&lt;p&gt;[3] Slunyaev, A., Klein, M., Clauss, G.F., Laboratory and numerical study of intense envelope solitons of water waves: generation, reflection from a wall and collisions. Physics of Fluids, 2017, Vol. 29, 047103.&lt;/p&gt;&lt;p&gt;[4] West, B.J., Brueckner, K.A., Janda, R.S., Milder, D.M., Milton, R.L., A new numerical method for surface hydrodynamics. J. Geophys. Res., 1987, Vol. 92, 11803-11824.&lt;/p&gt;&lt;p&gt;[5] Ducrozet, G., Gouin, M., Influence of varying bathymetry in rogue wave occurrence within unidirectional and directional sea-states. J. Ocean Eng. Mar. Energy, 2017, Vol. 3, 309-324.&lt;/p&gt;