Oscillatory integrals generated by the Ostrovsky equation

Dynamics of internal solitary waves in a rotating fluid

The large‐amplitude internal waves commonly observed in the coastal ocean often take the form of unsteady undular bores. Hence, here, we examine the long‐time combined effect of variable topography and background rotation on the propagation of internal undular bores, using the framework of a variable‐coefficient Ostrovsky equation. Because the leading waves in an internal undular bore are close to solitary waves, we first examine the evolution of a single solitary wave. Then, we consider an internal undular bore, for which two methods of generation are used. One method is the matured undular bore developed from an initial shock box in the Korteweg–de Vries equation, that is the Ostrovsky equation with the rotational term omitted, and the other method is a modulated cnoidal wave solution of the same Korteweg–de Vries equation. It transpires that in the long‐time model simulations, the rotational effect disintegrates the nonlinear waves into inertia‐gravity waves, and then there emerge complicated interactions between these inertia‐gravity waves and the modulated periodic waves of the undular bore, especially at the rear part of the undular bore. However, near the front of the undular bore, nonlinear effects further modulate these waves, with the eventual emergence of nonlinear envelope wave packets.

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

Solitary-wave families of the Ostrovsky equation: An approach via reversible systems theory and normal forms

This paper derives the Whitham modulation equations for the Ostrovsky equation. The equations are then used to analyse localized cnoidal wavepacket solutions of the Ostrovsky equation in the weak rotation limit. The analysis is split into two main parameter regimes: the Ostrovsky equation with normal dispersion relevant to typical oceanic parameters and the Ostrovsky equation with anomalous dispersion relevant to strongly sheared oceanic flows and other physical systems. For anomalous dispersion a new steady, symmetric cnoidal wavepacket solution is presented. The new wavepacket can be represented as a solution of the modulation equations and an analytical solution for the outer solution of the wavepacket is given. For normal dispersion the modulation equations are used to describe the unsteady finite-amplitude wavepacket solutions produced from the rotation-induced decay of a Korteweg-de Vries solitary wave. Again, an analytical solution for the outer solution can be given. The centre of the wavepacket closely approximates a train of solitary waves with the results suggesting that the unsteady wavepacket is a localized, modulated cnoidal wavetrain. The formation of wavepackets from solitary wave initial conditions is considered, contrasting the rapid formation of the packets in anomalous dispersion with the slower formation of unsteady packets under normal dispersion.

Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich’s $f$ -plane extension of the Miyata–Maltseva–Choi–Camassa model. We develop simultaneous asymptotic expansions for the evolving mean fields and deviations from them to describe a large class of uni-directional waves via the Ostrovsky equation, which fully decouples from mean-field variations. The latter generate additive inertial oscillations in the shear and in the phase of both the interfacial displacement and shear. Unlike conventional derivations leading to the Ostrovsky equation, our formulation does not impose the zero-mean constraints on the initial conditions of any variable. Using the constructed solutions, we model the evolution of quasi-periodic initial conditions close to the cnoidal wave solutions of the Korteweg–de Vries (KdV) equation but with local defects, both with and without rotation. We show that rotation leads to the emergence of bursts of internal waves and shear currents, qualitatively similar to the wavepackets generated from solitons and modulated cnoidal waves in earlier studies, but emerging much faster. We also show that cnoidal waves with expansion defects discussed in this work are generalised travelling waves of the KdV equation: they satisfy all conservation laws of the KdV equation (appropriately understood), as well as the Weirstrass–Erdmann corner condition for broken extremals of the associated variational problem and a natural weak formulation. Being smoothed in numerical simulations, they behave, in the absence of rotation, as long-lived states with no visible evolution, while rotation leads to the emergence of strong bursts.

In this work, the effect of variable medium is taken into account to the Ostrovsky equation, namely variable-coefficient Ostrovsky equation. Ostrovsky equation originally is obtained from Korteweg-de Vries equation with an additional linear non-local term representing the effects of background rotation. The addition of this term has the drastic effect of destroying the solitary wave solution. Instead an initial solitary-like disturbance decays into radiating oscillatory waves and eventually form a nonlinear wave packet. However, a uniform background state, it not sufficient to describe internal solitary waves in the ocean. So, the effect of variable medium on the propagation of oceanic internal solitary waves in the present of background rotation is studied. This effect alone to the Ostrovsky equation can change the size of amplitude and wave packet envelope.

This paper focuses on the study of linear and non-linear surface and internal waves, in a complete setting, using a two-layer model of a stratified fluid. The respective Korteweg-de Vries evolutionary equations have been obtained, analysed, and compared with the ‘rigid lid’ model data. Boussinesq-type equations have been derived for the interacting modes pertaining to one type and to different types. It is shown that in addition to the known mechanisms of interaction between internal and surface waves, interaction between long non-linear baroclinic modes and barotropic modes, propagating in the same direction, is likely in such a system.

In this chapter, we present some examples of nonlinear evolution equations in one space dimension. We re-discuss the traditional Korteweg–de Vries (KdV) equation for the shallow water long channel case, and its cnoidal waves and soliton solutions. Then we briefly present the MKdV equation and some nonlinear dispersion extension of it. In the last sections, we discuss some possible dynamical generalizations of the shallow water models on compact intervals, for any depth of the fluid. The resulting equation is an infinite-order differential one, and it reduces to a finite difference differential equation. We show that this generalized KdV equation approaches the KdV, MKdV, and Camassa–Holm limiting equations, both at the equation and at the solution level, in the appropriate physical conditions. In the last part we discuss the Boussinesq equations on a circle.

In this review we consider the propagation of nonlinear waves in magnetically structured plasmas. We start from studying nonlinear surface waves on magnetic interfaces and give a derivation of the equation governing propagation of these waves in incompressible plasmas. We present the results of a numerical solution of this equation. Then we discuss various generalizations of this equation.We proceed to study slow nonlinear waves in magnetic slabs. We present the Benjamin-Ono equation describing nonlinear slow sausage surface waves in magnetic slabs, and discuss its solutions in the form of algebraic solitons. Then we consider nonlinear slow body waves and write down the governing equation for these waves. We discuss its numerical solution which shows that the evolution of an initial perturbation results in a gradient catastrophe.After that we consider nonlinear slow sausage surface and slow body waves in magnetic tubes, which are similar to the corresponding counterparts in magnetic slabs. We present a system of two equations describing both types of waves. We briefly discuss the derivation of the Leibovich-Roberts equation for slow sausage surface waves from this system of equations, and present the results of its numerical study. Then we give a brief description of the derivation of the governing equation for slow body waves, which is very similar to its counterpart for slow body waves in magnetic slabs and has very similar properties. We complete this part of the review with presenting the governing equation for non-axisymmetric surface waves in a thin magnetic tube in an incompressible plasma.In the last part of the paper we consider the damping of nonlinear waves in weakly dissipative magnetically structured plasmas. We briefly discuss the application of the obtained results to the problem of heating of the solar corona.KeywordsSurface WaveSolitary WaveSolar PhysMagnetic TubeKink ModeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This work considers the solution of Cauchy problem (initial value problem) in a setting of arithmetic, algebra, and topology provided by Observer’s Mathematics (see www.mathrelativity.com) and applies this solution to free wave equation, the linear (time-dependent) Schrodinger equation, the (time-dependent) Airy equation, the Korteweg-de Vries (KdV) equation, and quantum theory of two-slit interference. Certain results and communications pertaining to these problems are provided.

The Korteweg-de Vries (KdV) equation is a mathematical model that describes the propagation of long waves in dispersive media. It takes into account both nonlinearity and dispersion, and is particularly useful for modeling phenomena like solitons. The nonlinear Schrödinger (NLS) equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. It is a useful model for describing many physical systems, including Bose-Einstein condensates, optical fibers, and water waves. A system that couples the KdV and NLS equations can model the interaction of long and short waves. This system combines the strengths of both models. The long waves described by the KdV equation can affect the behavior of the short waves described by the NLS equation, while the short waves can in turn affect the behavior of the long waves. Such a coupled system has been studied extensively over the last few decades, and has led to important insights into many physical systems. This paper considers the existence of local solutions to the Cauchy problem of KdV-Schrödinger nonlinear system on the basis of literature (Bernard D, Nghiem V N, Benjamin L S <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1088/1751-8113/49/41/415501">2016 <i>J. Phys. A: Math. Theor.</i> <b>49</b> 415501</ext-link>), and also gives the existence space of the local solutions.

Modelling and observations of oceanic nonlinear internal wave packets affected by the Earth’s rotation

The authors investigate the propagation of nonlinear high frequency TM surface waves on a thin unmagnetized plasma layer bounded by vacuum. It is found that envelope surface wave solitons from the modulation of finite amplitude electron waves by slow ion acoustic motion exist. The non-existence of solitons with purely electronic modulation is also discussed.

The propagation of monochromatic nonlinear symmetric hybrid surface waves in a cylindrical nonlinear anisotropic inhomogeneous metal-dielectric waveguide is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations. Spectral parameters of the problem are propagation constants of the waveguide. The setting is reduced to a new type of nonlinear eigenvalue problem and an analytical method of its solution is developed. The propagation modes are found that have not been previously reported in the literature. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.