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A combination of multiple data sources has been used to study the characteristics of internal solitary waves (ISWs) in the Gulf of Oman (GoO). Water column stratification in the Gulf has been examined using field observations and World Ocean Atlas 2018 datasets. The spatiotemporal distribution of ISWs has been obtained from satellite images obtained by means of Synthetic Aperture Radar (SAR) and optical sensors taken from 2018 to 2020. The mechanisms of ISW generation in the GoO have been studied using the data revealed from different available sources. The results show that there are annually two major typical stratifications in the GoO throughout the year, strong stratification in May through September and weak stratification during other months. Dispersion relations corresponding to these types of stratification have been obtained with acceptable accuracy for both deep and shallow regions. The spatiotemporal distribution of ISWs demonstrates that the western and southern regions of the GoO are the hotspots for generation of ISWs in this basin. Several mechanisms of ISW generation in the GoO are discussed including tide, eddies, lee waves, and atmospheric perturbation; the latter one is, apparently, responsible for the appearance of large-amplitude ISWs.

A simple kinematic approach to the description of the interaction between solitons is developed. It is applicable to both integrable and non-integrable two-dimensional models, including those commonly used for studying the surface and internal oceanic waves. This approach allows obtaining some important characteristics of the interaction between solitary waves propagating at an angle to each other. The developed theory is validated by comparison with the exact solutions of the Kadomtsev-Petviashvili equation and then applied to the observed interaction of solitary internal waves in a two-layer fluid within the two-dimensional Benjamin-Ono model.

We study the self-modulation of flexural-gravity waves on a water surface covered by a compressed ice sheet. For weakly nonlinear long perturbations of the potential flow, we derive the nonlinear Schrödinger equation and investigate the conditions when a quasi-sinusoidal wave becomes unstable with respect to the amplitude modulation. The domains of instability are presented in the planes of parameters depending on the values of the local water depth, and the coefficients of ice rigidity/elasticity and of longitudinal stress. We show that under some conditions the occurrence of the modulational instability of oceanic waves under ice looks feasible and present estimates for the real oceanic conditions. Depending on the conditions, bright envelope solitons or dark solitons can emerge on the surface.A.S. acknowledges the support from Laboratory of Dynamical Systems and Applications NRU HSE (the Ministry of Science and Higher Education of the Russian Federation Grant No. 075-15-2019-1931) and by the Russian Foundation for Basic Research (Grant No. 21-55-15008). Y.S. acknowledges the funding provided by the grant No. FSWE-2020-0007 through the State task program in the sphere of scientific activity of the Ministry of Science and Higher Education of the Russian Federation.

As it is known, the problem of finding traveling waves in 1D nonlinear and dispersive media may be reduced to solving a system of ordinary differential equations. If the order of the system is large, then the internal wave structure can be very complicated and even random. If the medium is inhomogeneous, it is natural to expect the absence of solutions in the form of traveling waves due to the reflection and multiple reflection effects. If, however, the medium parameters change slowly (in comparison to the wavelength), and the reflection is weak, it becomes possible to construct an approximate solution in the form of a traveling wave with a variable amplitude and phase by using asymptotic methods (WKB, geometric optics or acoustics). For the media with a monotonic change in parameters, such solutions demonstrate the highest gain and the ability to transmit a signal over long distances without distortion.It turns out to be possible to find exact solutions in the form of traveling waves with variable amplitude and phase in highly inhomogeneous media under certain assumptions on the medium parameters. Our paper reviews possible approaches to finding travelling reflectionless waves in the shallow water channels with variable cross-sections and currents. The basic equations are the classical 1D nonlinear shallow-water equations for water displacement and velocity averaged through the cross-channel. Mathematical procedure to get the solutions in the form of travelling reflectionless waves is based on the transformation of the original equations with variable coefficients to the constant-coefficient PDE. We first demonstrate this procedure using the example of the classical linear wave equation with variable coefficient when it can be reduced to the Klein-Gordon equation with constant coefficients. This gives rise to an ordinary second-order differential equation for finding a variable coefficient (the wave speed), so that traveling waves exist in a wide class of inhomogeneous water channels. The second procedure is the reducing of variable-coefficient 1D wave equation to the spherical symmetric wave equation in the odd-dimensional space, where waves traveling to and from the center are separated. More complicated procedure is developed for the channels with non-uniform current. In conclusion, we discuss the effectiveness of this procedure in the framework of Boussinesq systems.The study is supported by grants RFBR (20-05-00162, 21-55-15008, 19-35-60022), President of the RF for the state support of Leading Scientific Schools of the RF (Grant No. NSH-70.2022.1.5).Recent publications:Didenkulova I. and Pelinovsky E. On shallow water rogue wave formation in strongly inhomogeneous channels. Journal of Physics A: Mathematical and Theoretical, 2016, vol. 49, 194001. Pelinovsky E., Didenkulova I., Shurgalina E., and Aseeva N. Nonlinear wave dynamics in self-consistent water channels. J Phys. A, 2017, vol. 50, 505501. Pelinovsky E., Talipova T., Didenkulova I., Didenkulova E. Interfacial long traveling waves in a two-layer fluid with variable depth. Studies in Applied Mathematics, 2019, vol. 142, No. 4, 513–527. Churilov S.M., Stepanyants Yu.A. Reflectionless wave propagation on shallow water with variable bathymetry and current. J. Fluid Mech., 2022, vol. 931, A15.

We show that in the linear approximation there are three classes of reflectionless wave propagation on a surface of shallow water in the channel with spatially varying depth, width and current speed. Two of these classes have been described in our previous paper (Churilov & Stepanyants, J. Fluid Mech., vol. 931, 2022, A15), and the third one was discovered recently and is described here. The general analysis of the problem shows that, within the approach used in both of our papers, these three classes apparently exhaust all possible cases of exact solutions of the problem considered. We show that the reflectionless flow can be global for certain conditions, i.e. it can exist on the entire $x$ -axis. There are also reflectionless flows which exist only on limited intervals of the $x$ -axis.

In the linear approximation, we study a one-dimensional problem of the reflectionless wave propagation on a surface of a shallow duct with the spatially varying water depth, duct width, and current. We show that both global and bounded exact solutions describing reflectionless propagation in opposite directions of long waves of arbitrary shape exist for the particular variations of these parameters. A general analysis of the problem is illustrated by a few solutions constructed for the specific cases of spatial profiles of the flow velocity. The results obtained can be of interest to mitigate the possible impact of waves on ships, marine engineering constructions, and human activity in the coastal zones., 25 pages, 10 figures

An interesting physical phenomenon was recently observed when a fresh-water basin is covered by a thin ice film that has properties similar to that of a rubber membrane. Surface waves can be generated under the action of wind on the air-water interface that contains an ice film. The modulation property of hydro-elastic waves (HEWs) in deep water covered by thin ice film blown by the wind with a uniform vertical profile is studied here in terms of the air-flow velocity versus a wavenumber. The modulation instability of HEWs is studied through the analysis of coefficients of the nonlinear Schrödinger (NLS) equation with the help of the Lighthill criterion. The NLS equation is derived using the multiple scale method in the presence of airflow. It is demonstrated that the potentially unstable hydro-elastic waves with negative energy appear for relatively small wind speeds, whereas the Kelvin–Helmholtz instability arises when the wind speed becomes fairly strong. Estimates of parameters of modulated waves for the typical conditions are given.

As it was shown earlier, a wide class of nonlinear 3-dimensional (3D) fluid flows of incompressible viscous fluid can be described by only one scalar function dubbed the quasi-potential. This class of fluid flows is characterized by a three-component velocity field having a two-component vorticity field. Both these fields may, in general, depend on all three spatial variables and time. In this paper, the governing equations for the quasi-potential are derived and simple illustrative examples of 3D flows in the Cartesian coordinates are presented. The generalisation of the developed approach to the fluid flows in the cylindrical and spherical coordinate frames represents a nontrivial problem that has not been solved yet. In this paper, this gap is filled and the concept of a quasi-potential to the cylindrical and spherical coordinate frames is further developed. A few illustrative examples are presented which can be of interest for practical applications.

This review presents theoretical, numerical, and experimental results of a study of the structure and dynamics of weakly nonlinear internal waves in a rotating ocean accumulated over the past 40 years since the time when the approximate equation, called the Ostrovsky equation, was derived in 1978. The relationship of this equation with other well-known wave equations, the integrability of the Ostrovsky equation, and the condition for the existence of stationary solitary waves and envelope solitary waves are discussed. The adiabatic dynamics of Korteweg–de Vries solitons in the presence of fluid rotation is described. The mutual influence of the ocean inhomogeneity and rotation effect on the dynamics of solitary waves is considered. The universality of the Ostrovsky equation as applied to waves in other media (solids, plasma, quark–gluon plasma, and optics) is noted.

We consider a transformation of long linear waves in an ocean with a variable depth. We calculate the transformation coefficients (the coefficients of transmission and reflection) as the functions of frequency and the total depth drop for three typical models of bottom profile variation: (i) piecewise-linear, (ii) piecewise-quadratic, and (iii) hyperbolic tangent profiles. For all these cases, exact solutions are obtained, analysed and graphically illustrated. This allows us to derive the transformation coefficients in the analytic form for the subsequent comparison with the results of approximate, numerical, or experimental study. We show that the results obtained are in agreement with both the energy flux conservation and Lamb’s formulae in the limiting case of zero frequency. We also study wave transformation on the underwater barriers and trenches of different shapes and compare the results obtained.

Using the Hirota bilinear method, we derive resonant solutions to the KP1 equation. Solutions describe lump chains differently oriented in (x, y)-plane. We show that resonant solutions arise as the limiting case of more general non-resonant solutions when phase shifts of lump chains caused by their interaction become infinite. Resonant solutions can describe both stationary patterns (for example, Y-shaped patterns consisting of three different lump chains) and non-stationary interacting parallel lump chains. In the latter case, a lump chain can be emitted/absorbed by another lump chain. As the number of lump chains increases, resonance phenomena become more complex and diversified including the cases of exchange of a lump chain by two other lump chains. The method used in this paper can be extended to apply to other integrable systems in two and three spatial dimensions such as, for example, described by Mel’nikov’s equations.

The modulation instability of surface capillary-gravity water waves is analysed in a shear flow model with a tangential discontinuity of velocity. It is assumed that air blows along the surface of the water with a uniform profile in the vertical direction. Such a model, despite its simplicity, plays an important role in hydrodynamics as the reference model for investigating basic physical phenomena of wave–current interactions and acquiring insights into a series of complex phenomena. In certain cases where the wavelength of interfacial perturbations is much bigger than the width of the shear flow profile, the model with the tangential discontinuity in the velocity is adequate for describing physical phenomena at least within limited spatial and temporal frameworks. A detailed analysis of the air-flow conditions under which modulation instability sets in is presented. It is also shown that the interfacial waves are subject to dissipative or radiative instability when negative-energy waves appear at the interface.

This paper presents the calculation of the hydrodynamic forces exerted on an oscillating circular cylinder when it moves perpendicular to its axis in infinitely deep water covered by compressed ice. The cylinder can oscillate both horizontally and vertically in the course of its translational motion. In the linear approximation, a solution is found for the steady wave motion generated by the cylinder within the hydrodynamic set of equations for the incompressible ideal fluid. It is shown that, depending on the rate of ice compression, both normal and anomalous dispersion can occur in the system. In the latter case, the group velocity can be opposite to the phase velocity in a certain range of wavenumbers. The dependences of the hydrodynamic loads exerted on the cylinder (the added mass, damping coefficients, wave resistance and lift force) on the translational velocity and frequency of oscillation were studied. It was shown that there is a possibility of the appearance of negative values for the damping coefficients at the relatively big cylinder velocity, then, the wave resistance decreases with the increase in cylinder velocity. The theoretical results were underpinned by the numerical calculations for the real parameters of ice and cylinder motion.

We study hydrodynamic models of astrophysical wormholes when water waves can be amplified in the course of propagation between two critical points where wave and current speeds coincide. Such models can be realised in shallow laboratory ducts with variable width and depths. In this paper, we derive the basic set of equations for shallow water waves on the spatially variable flow in the duct of a variable cross-section and present the asymptotic analysis of solutions in the neighbourhood of the critical points. The critical points mimic either the black hole (BH) horizon if the flow transits from the subcritical to the supercritical regime, or the white hole (WH) horizon if the flow transits from the supercritical to the subcritical regime. We study then, the wave propagation in the flow with two horizons when the flow transits first the BH horizon and then the WH one. The region between the horizons mimics a wormhole in general relativity. For the sake of completeness, we also study the successive transition through WH and BH horizons. The theoretical results are illustrated by numerical calculations of wave propagation in both these arrangements. It is shown that the wave amplification after passing the active zone between the horizons takes place in BH{WH arrangements only and can occur for different relationships between the subcritical and supercritical flow velocities. The frequency dependence of the amplification factor is obtained and quantified in terms of the velocity ratio within and outside the \wormhole domain". In the next paper, we plan to present exact solutions for the specific velocity profile., Comment: 49 pages, 18 figures, 2 tables

In this paper we study the transformation of surface envelope solitons travelling over a bottom step in water of a finite depth. Using the transformation coefficients earlier derived in the linear approximation, we find the parameters of transmitted pulses and subsequent evolution of the pulses in the course of propagation. Relying on the weakly nonlinear theory, the analytic formulae are derived which describe the maximum attainable wave amplitude in the neighbourhood of the step and in the far zone. Solitary waves may be greatly amplified (within the weakly nonlinear theory formally even without a limit) when propagating from relatively shallow water to the deeper domain due to the constructive interference between the newly emerging envelope solitons and the residual quasi-linear waves. The theoretical results are in a good agreement with the data of direct numerical modelling of soliton transformation. In particular, more than double wave amplification is demonstrated in the performed simulations.

A simple kinematic approach to the description of interaction between solitons is developed. It is applicable to both integrable and non-integrable two-dimensional models, including those commonly used for studying surface and internal oceanic waves. This approach allows for obtaining some important characteristics of the interaction between solitary waves propagating at an angle to each other. The developed theory is validated by comparison with the exact solutions of the Kadomtsev–Petviashvili equation and then applied to the observed interaction of solitary internal waves in a two-layer fluid within the two-dimensional Benjamin–Ono model.

In the linear approximation, we study wave motions of a compressed elastic ice sheet caused by the motion of a two-dimensional dipole in the water beneath the sheet. The fluid flow is described by the potential theory, while the ice sheet is modelled through a thin elastic plate floating on the water surface. The solution for the vertical displacement of the ice sheet is derived for a transient dipole undergoing arbitrary two-dimensional motion. Three cases are considered in detail when the dipole moves horizontally with a uniform speed at some depth or horizontally oscillates, or moves and oscillates. The formulae for the plate displacement are derived for the fluid of finite depth, but then analysed in detail for the infinitely deep case. We show that the character of the solutions is different in the different domains of the parameter plane and classify the possible cases. Then we calculate the wave patterns on the plate for the different regimes of dipole motion and typical values of plate parameters. The studied problem can be considered as the simplified model of motion of a circular cylinder in a water under an ice cover. In the last section we compare the characteristics of wave motions onsetting in the far-field zone of the flow around a circular cylinder and its dipole approximation and show that the difference in the wave characteristics and force loads for these two cases is small and quickly vanishes when the ice plate thickness increases. In conclusion, we present estimates of amplitudes and wavelengths of wave perturbations for the real oceanic conditions.

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 Schrödinger 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].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 h2/h1, where h1 and h2 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.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.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.[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, 214–223.[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.[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.[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.[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.

The interactions of multi-lumps within the Kadomtsev–Petviashvili-1 (KP1) equation are studied analytically and numerically. The dependence of stationary multi-lump structures on free parameters is discussed. The interactions of single lumps with each other and with a more complex objects such as bi-lumps, as well as the interactions of bi-lumps with each other are studied numerically. The results obtained are described and illustrated graphically (the videos are also available).

Complex dynamics of small particles under the action of acoustic radiation force is considered. This work extends the previous studies of Ostrovsky & Sarvazyan (2009) and Ostrovsky (2015) in two aspects. Firstly, here the particle material can have an arbitrary compressibility and sound speed. Secondly, in addition to the viscosity effect described by the Stokes drag force, the Boussinesq–Basset drag force and the inertial force which includes added mass effect are accounted for. Although, as was assumed in the earlier publications, the latter effects are usually small, however their influence can be noticeable in specific cases considered in this paper. The control of particle motion by switching of acoustic modes is also studied for particles of different properties. Quantitative estimates are given for particles made of different materials.

We consider the dynamics of internal envelope solitons in a two-layer rotating fluid with a linearly varying bottom. It is shown that the most probable frequency of a carrier wave which constitutes the solitary wave is the frequency where the growth rate of modulation instability is maximal. An envelope solitary wave of this frequency can be described by the conventional nonlinear Schr&ouml, dinger equation. A soliton solution to this equation is presented for the time-like version of the nonlinear Schr&ouml, dinger equation. When such an envelope soliton enters a coastal zone where the bottom gradually linearly increases, then it experiences an adiabatical transformation. This leads to an increase in soliton amplitude, velocity, and period of a carrier wave, whereas its duration decreases. It is shown that the soliton becomes taller and narrower. At some distance it looks like a breather, a narrow non-stationary solitary wave. The dependences of the soliton parameters on the distance when it moves towards the shoaling are found from the conservation laws and analysed graphically. Estimates for the real ocean are presented.

The interaction of a solitary wave with a long background periodic wave in a deep ocean is studied within the framework of the rotation modified Benjamin–Ono equation. Using a multiple scales analysis, we find stationary and nonstationary solutions for a Benjamin–Ono soliton trapped within a long sinusoidal wave. We show that the radiation losses experienced by the soliton and caused by the Earth’s rotation can be compensated by energy pumping from the background wave. However, the feedback of the soliton on the background wave eventually leads to the destruction of the coherent structure and energy dispersion in a quasi-random wave field. Estimates of the characteristic parameters of the soliton dynamics are presented for real oceanic conditions.

Interaction of a soliton with long background wave s is studied within the framework of rotation modified Korteweg–de Vries (rKdV) equation. Using the asymptotic method for solitons propagating in the field of a long background wave we derive a set of ODEs describing soliton amplitude and phase with respect to the background wave. The shape of the background wave may range from a sinusoid to the limiting profile representing a periodic sequence of parabolic arcs. We analyse energy exchange between a soliton and the long wave taking radiation losses into account. It is shown that the losses can be compensated by energy pumping from the long wave and, as the result, a stationary soliton can exist, unlike the case when there is no variable background. A more complex case when a free long wave attenuates due to the energy consumption by a soliton is also considered. Some of the analytical results are compared with the results of direct numerical calculations within the framework of the rKdV equation.

Obliquely propagating skew lumps are studied within the framework of the Kadomtsev–Petviashvili equation with a positive dispersion (the KP1 equation). Specific features of such lumps are analysed in detail. It is shown that skew multi-lump solutions can be also constructed within the framework of such an equation. As an example, the bi-lump solution is presented in the explicit form, analysed and illustrated graphically. The relevance of skew lumps to the real physical systems is discussed.

We present a derivation of the time averaged potential and kinetic energies for small-amplitude surface waves on a shear flow with constant vorticity. The effect of surface tension is also taken into consideration. It is demonstrated that the virial theorem of the energy equipartition between the potential and kinetic components is not valid in general for waves on a shear flow. We also show that waves with a negative energy may exist in a shear flow, and we find the condition for existence of such waves.

We present an example of systematic use of asymptotic methods for the description of long waves at the interface between two fluids of different densities. The governing equations for weakly nonlinear long interfacial waves are consistently derived for a general case of nonpotential flow and taking into account surface tension between two layers. Particular attention is given to the situations when some terms in the resulting fifth order KdV-type evolution equation become small, vanish, or change their sign.

We study weakly nonlinear wave perturbations propagating in a cold nonrelativistic and magnetized ideal quark-gluon plasma. We show that such perturbations can be described by the Ostrovsky equation. The derivation of this equation is presented for the baryon density perturbations. Then we show that the generalized nonlinear Schr{��}dinger (NLS) equation can be derived from the Ostrovsky equation for the description of quasi-harmonic wave trains. This equation is modulationally stable for the wave number $k < k_m$ and unstable for $k > k_m$, where $k_m$ is the wave number where the group velocity has a maximum. We study numerically the dynamics of initial wave packets with the different carrier wave numbers and demonstrate that depending on the initial parameters they can evolve either into the NLS envelope solitons or into dispersive wave trains.

In the linear approximation we study long wave scattering on an axially symmetric flow in a shallow water basin with a drain in the center. This classical problem can be considered as a model of wave scattering on a rotating black hole. For the low frequencies analytic solutions are derived to describe both pure potential perturbations (surface gravity waves), and perturbations with nonzero potential vorticity. For moderate frequencies such solutions are obtained numerically and illustrated graphically. It is shown that there are two processes governing the dynamics of perturbations, namely, the scattering of incident gravity waves by central vortex, and the emission of gravity waves stimulated by the potential vorticity. Some aspects of their joint action are discussed., 32 pages, 5 figures

This paper presents specific features of solitary wave dynamics within the framework of the Ostrovsky equation with variable coefficients in relation to surface and internal waves in a rotating ocean with a variable bottom topography. For solitary waves moving toward the beach, the terminal decay caused by the rotation effect can be suppressed by the shoaling effect. Two basic examples of a bottom profile are analyzed in detail and supported by direct numerical modelling. One of them is a constant-slope bottom and the other is a specific bottom profile providing a constant amplitude solitary wave. Estimates with real oceanic parameters show that the predicted effects of stable soliton dynamics in a coastal zone can occur, in particular, for internal waves., 31 pages, 17 figures

A theoretical basis for the possibility to emulate Bose–Einstein condensates (BEC) in a water tank is elaborated. It is shown that the equation governing the mean-field BEC dynamics, viz., the Gross–Pitaevskii (GP) equation, can be derived in the same form for surface water waves in a tank with a spatially varying background current. Depending on the wavenumber of the carrier wave, the GP equation for the surface waves is tantamount to the GP equation corresponding to the attractive or repulsive inter-atomic interactions in BEC. The external potential in the effective GP equation can be easily implemented in the water tank with an uneven bottom featuring a well or hump. Examples of particular stable exact solutions of the GP equation with the effective potential are presented. Estimates for physical parameters of the hydrodynamic setting are given.

It has been shown in our previous paper that there is a wide class of 3D motions of incompressible viscous fluid which can be described by one scalar function dabbed the quasi-potential. This class of fluid flows is characterized by three-component velocity field having two-component vorticity field; both these fields can depend of all three spatial variables and time, in general. Governing equations for the quasi-potential have been derived and simple illustrative example of 3D flow has been presented. Here, we derive the Bernoulli integral for that class of flows and compare it against the known Bernoulli integrals for the potential flows or 2D stationary vortical flows of inviscid fluid. We show that the Bernoulli integral for this class of fluid motion possesses unusual features: it is valid for the vortical nonstationary motions of a viscous incompressible fluid. We present a new very nontrivial analytical example of 3D flow with two-component vorticity which hardly can be obtained by any of known methods. In the last section, we suggest a generalization of the developed concept which allows one to describe a certain class of 3D flows with the 3D vorticity.

A new model equation describing weakly nonlinear long internal waves at the interface between two thin layers of different density is derived for the specific relationships between the densities, layer thicknesses and surface tension between the layers. The equation derived and dubbed here the Gardner–Kawahara equation represents a natural generalisation of the well-known Korteweg–de Vries (KdV) equation containing the cubic nonlinear term as well as fifth-order dispersion term. Solitary wave solutions are investigated numerically and categorised in terms of two dimensionless parameters, the wave speed and fifth-order dispersion. The equation derived may be applicable to wave description in other media.

Flexural transverse waves in an anharmonic chain of atoms is considered and the nonlinear vector equation for the phonon modes in the long-wave approximation is derived taking into account the weak dispersion effects. Particular cases of the equation derived are discussed; among them the vector mKdV equation (Gorbacheva and Ostrovsky, 1983) [12], as well as the new model vector equations dubbed here the ‘second order cubic Benjamin–Ono (socBO) equation’ and ‘nonlinear pseudo-diffusion equation’. Stationary solutions to the equation derived are studied and it is found in which cases physically reasonable periodic and solitary type solutions may exist. The simplest non-stationary interactions of solitary waves of different polarisation are studied by means of numerical simulation. A new interesting phenomenon is revealed when two solitons of the same or opposite polarities interact elastically, whereas the interaction of two solitons lying initially in the perpendicular planes is essentially inelastic resulting in the survival of only one soliton and destruction of another one.

Flexural transverse waves in an anharmonic chain of atoms is considered and the nonlinear vector equation for the phonon modes in the long-wave approximation is derived taking into account the weak dispersion effects. Particular cases of the equation derived are discussed; among them the vector mKdV equation (Gorbacheva and Ostrovsky, 1983) [12], as well as the new model vector equations dubbed here the ‘second order cubic Benjamin–Ono (socBO) equation’ and ‘nonlinear pseudo-diffusion equation’. Stationary solutions to the equation derived are studied and it is found in which cases physically reasonable periodic and solitary type solutions may exist. The simplest non-stationary interactions of solitary waves of different polarisation are studied by means of numerical simulation. A new interesting phenomenon is revealed when two solitons of the same or opposite polarities interact elastically, whereas the interaction of two solitons lying initially in the perpendicular planes is essentially inelastic resulting in the survival of only one soliton and destruction of another one.

The decay of Kadomtsev - Petviashvili lumps is considered for a few typical dissipations - Rayleigh dissipation, Reynolds dissipation, Landau damping, Chezy bottom friction, viscous dissipation in the laminar boundary layer, and radiative losses caused by large-scale dispersion. It is shown that the straight-line motion of lumps is unstable under the influence of dissipation. The lump trajectories are calculated for two most typical models of dissipation - the Rayleigh and Reynolds dissipations. A comparison of analytical results obtained within the framework of asymptotic theory with the direct numerical calculations of the Kadomtsev - Petviashvili equation is presented. Good agreement between the theoretical and numerical results is obtained., 21 pages, 7 figures

The adiabatic decay of different types of internal wave solitons caused by the Earth rotation is studied within the framework of the Gardner-Ostrovsky equation. The governing equation describing such processes includes quadratic and cubic nonlinear terms, as well as the Boussinesq and Coriolis dispersions. It is shown that at the early stage of evolution solitons gradually decay under the influence of weak Earth rotation. The characteristic decay time is derived for different types of solitons for positive and negative coefficient of cubic nonlinearity (both signs of that parameter may occur in the oceans). The coefficient of quadratic nonlinearity determines only a polarity of solitary wave. It is found that the adiabatic theory describes well the decay of solitons having bell-shaped profiles. In contrast to that, large amplitude table-top solitons, which can exist when the cubic nonlinear term is negative, are structurally unstable. Under the influence of Earth rotation they transfer first to the bell-shaped solitons, which decay then adiabatically. Estimates of the characteristic decay time of internal solitons are presented for the real oceanographic conditions., 32 pages, 14 figures

We study solitary wave solutions of the fifth-order Korteweg - de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear dispersion, as well as two nonlinear dispersive terms. An exact solitary wave solution to this equation is derived and the dependence of its amplitude, width and speed on the parameters of the governing equation are studied. It is shown that the derived solution can represent either an embedded or regular soliton depending on the equation parameters. The nonlinear dispersive terms can drastically influence the existence of solitary waves, their nature (regular or embedded), profile, polarity, and stability with respect to small perturbations. We show, in particular, that in some cases embedded solitons can be stable even with respect to interactions with regular solitons. The results obtained are applicable to surface and internal waves in fluids, as well as to waves in other media (plasma, solid waveguides, elastic media with microstructure, etc.)., Comment: 37 pages, 27 figures