Ostrovsky Equation Research Articles

Modulational Instability in the Ostrovsky Equation and Related Models

Modulational Instability in the Ostrovsky Equation and Related Models

Mathematical analysis of a norm-conservative numerical scheme for the Ostrovsky equation

AbstractThe target of this study is a norm-conservative scheme for the Ostrovsky equation, as its mathematical analysis has not been addressed. First, the existence and uniqueness of its numerical solutions are demonstrated. Subsequently, a convergence estimate in the two-norm is established. This, in turn, implies a convergence in the first-order Sobolev space using a supplementary sup-norm boundedness argument. Finally, this conservative scheme can be implemented in a differential form, which is considerably better than the integral form in terms of computational cost-effectiveness.

The local well-posedness of the coupled Ostrovsky system with low regularity

In this paper, the Cauchy problem for the coupled Ostrovsky equations with an initial value in the Sobolev spaces Hs(R)×Hs(R) of lower order s is considered. With the bilinear estimate, it is proved that the initial value problem is locally well-posed in Hs(R)×Hs(R) for s>−34 by using Bourgain spaces. Moreover, if s<−34, it is demonstrated that one of the nonlinear iteration from the initial data to the putative solutions is discontinuous with an argument on the high-to-low frequency. In this sense, it is then concluded that the coupled Ostrovsky equations is ill-posed in Hs(R)×Hs(R) for s<−34.

Investigating invariance and conservation laws for the classes of nonlinear parabolic and wave systems

We study various classes of the nonlinear dynamics of some high order parabolic equations like the Oskolkov–Benjamin–Bona–Mahony–Burgers and Benjamin–Bona–Mahony–Peregrine–Burger equations that arise in the study of some wave phenomena. Also, a broader class of partial differential equations are used in modelling ocean waves that originate from the Ostrovsky equation. We study the invariance properties via the Lie invariance method for the nonlinear systems and further establish classes of conservation laws for models arises in this study. We show how the relationship leads to double reductions of the systems. This relationship is determined by a recent result involving multipliers that lead to a total divergence or closed form of the differential equation under investigation.

Detecting delamination via nonlinear wave scattering in a bonded elastic bar

In this paper we examine the effect of delamination on wave scattering, with the aim of creating a control measure for layered waveguides of various bonding types. Previous works have considered specific widths of solitary waves for the simulations, without analysing the effect of changing the soliton parameters. We consider two multi-layered structures: one containing delamination ‘sandwiched’ by perfect bonding and one containing delamination but ‘sandwiched’ by soft bonding. These structures are modelled by coupled Boussinesq-type equations. Matched asymptotic multiple-scale expansions lead to coupled Ostrovsky equations in soft bonded regions and Korteweg-de Vries equations in the perfectly bonded and delaminated region. We use the Inverse Scattering Transform to predict the behaviour in the delaminated regions. In both cases, numerical analysis shows that we can predict the delamination length by changes in the wave structure, and that these changes depend upon the Full Width at Half Magnitude (FWHM) of the incident soliton. In the case of perfect bonding, we derive a theoretical prediction for the change and confirm this numerically. For the soft bonding case, we numerically identify a similar relationship using the change in amplitude. Therefore we only need to compute one curve to determine the behaviour for any incident solitary wave, creating a framework for designing measurement campaigns for rigorously testing the integrity of layered structures.

A Regularization-Free Scheme for Recovering Large External Forces of Higher-Order Nonlinear Evolution Equations

In this study, the inverse engineering problems of the Ostrovsky equation (OE), Kawahara equation (KE), modified Kawahara equation (mKE), and sixth-order Korteweg-de Vries (KdV) equation will be investigated numerically. An effective numerical approach to tackle these inverse Coriolis dispersion problems and the above-mentioned inverse problems are still not available. To use different boundary shape functions, we must deal with the boundary data, initial conditions, and terminal time conditions of the OE, KE, mKE, and sixth-order KdV equations. The unknown Coriolis dispersion of OE and unknown large external forces of those three equations can be retrieved through back-substitution of the solution into the OE, KE, mKE, and sixth-order KdV equations while we obtain the solution with the symmetry property by employing the boundary shape function scheme (BSFS). Five numerical experiments with noisy data are carefully validated and discussed.

Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction.

Coupled Boussinesq equations are used to describe long weakly nonlinear longitudinal strain waves in a bi-layer with soft bonding between the layers (e.g., a soft adhesive). From a mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations for the right- and left-propagating waves in each layer. However, the models impose a "zero-mass constraint"; i.e., the initial conditions should necessarily have zero mean, restricting the applicability of that description. Here, we bypass the contradiction in this high-contrast case by constructing the solution for the deviation from the evolving mean value, using asymptotic multiple-scale expansions involving two pairs of fast characteristic variables and two slow time variables. By construction, the Ostrovsky equations emerging within the scope of this derivation are solved for initial conditions with zero mean, while initial conditions for the original system may have non-zero mean values. Asymptotic validity of the solution is carefully examined numerically. We apply the models to the description of counter-propagating waves generated by solitary wave initial conditions, or co-propagating waves generated by cnoidal wave initial conditions, as well as the resulting wave interactions, and contrast with the behavior of the waves in bi-layers when the linear long-wave speeds in the layers are close (low-contrast case). One local (classical) and two non-local (generalized) conservation laws of the coupled Boussinesq equations for strains are derived and used to control the accuracy of the numerical simulations.

On the radius of spatial analyticity for Ostrovsky equation with positive dispersion

ABSTRACT It is shown that the uniform radius of spatial analyticity of solutions at time t to the Ostrovsky equation with positive dispersion cannot decay faster than as given initial data that is analytic with fixed radius . The main ingredients in the proof are almost conservation law for the solution to the Ostrovsky equation in space of analytic functions and space-time dyadic bilinear estimates associated with the Ostrovsky equation.

The nonlinear evolution of internal tides. Part 1: the superharmonic cascade

In non-uniform stratification, horizontally propagating internal waves with the vertical structure of a single mode self-interact to excite superharmonics. Baker &amp; Sutherland (J. Fluid Mech., vol. 891, 2020, R1) showed that a vertical mode-1 parent wave of sufficiently small amplitude dominantly excites a vertical mode-1 superharmonic with double the horizontal wavenumber. Through theory, assuming a parent wave of sufficiently small amplitude, they showed that the superharmonics grew and decayed periodically due to the parent forcing frequency being off-resonant with the natural frequency of the superharmonic. Here, we extend this theory to allow for larger parent wave amplitudes and/or stronger resonant forcing, as would occur at lower latitudes, where the influence of background rotation is small. The resulting coupled system of nonlinear ordinary differential equations is shown to well predict the evolution of the internal tide as determined in fully nonlinear numerical simulations. With strong nonlinear forcing, successive superharmonics grow to non-negligible amplitudes in what we call the ‘superharmonic cascade’. The phase relationship between the superharmonics is such that when superimposed, the internal tide transforms into a solitary wave train, consistent with the predictions of well-established shallow-water models, particularly that of the Ostrovsky equation, which is an extension of the Korteweg–de Vries equation accounting for background rotation. This work thus gives new insight into internal solitary wave generation. The model equations have less restrictive assumptions than models based upon shallow-water theory, and because they are quickly solved, these provide a potentially powerful new tool to examine the nonlinear evolution of the internal tide.

Convergence problem of reduced Ostrovsky equation in Fourier–Lebesgue spaces with rough data and random data

This paper is devoted to studying the convergence problem of free reduced Ostrovsky equation in Fourier–Lebesgue spaces with rough data and the stochastic continuity of free reduced Ostrovsky equation in Fourier–Lebesgue spaces with random data. On the one hand, we establish the pointwise convergence related to the free reduced Ostrovsky equation in Fourier–Lebesgue spaces [Formula: see text] with rough data. In particular, we show that [Formula: see text] is the necessary condition for the maximal function estimate in [Formula: see text], which means that [Formula: see text] is optimal for rough data. On the other hand, we present the stochastic continuity of free reduced Ostrovsky equation at [Formula: see text] in Fourier–Lebesgue spaces [Formula: see text] with random data.

Scattering of an Ostrovsky wave packet in a delaminated waveguide

We examine the scattering of an Ostrovsky wave packet, generated from a solitary wave, in a two layered waveguide with a delamination in the centre and soft (imperfect) bonding either side of the centre. The lower layer of the waveguide is assumed to be significantly denser than the upper layer, leading to a system of Boussinesq–Klein–Gordon (BKG) equations. Direct numerical modelling is difficult and so a semi-analytical approach consisting of several matched asymptotic multiple-scale expansions is used, which leads to Ostrovsky equations in soft bonded regions and Korteweg–de Vries equations in the delaminated region. The semi-analytical approach and direct numerical simulations are in good agreement with each other and theoretical estimates. The dispersion relations are used to estimate the wave speed and hence classify the length of the delamination, in addition to changes in the amplitude of the wave packet. We also show how to scale the non-dimensional results to material variables and an example for PMMA is presented. These results can provide a tool to control the integrity of layered structures.

Axisymmetric longitudinal waves in a cylindrical shell interacting with a nonlinear elastic medium

The evolution of axisymmetric longitudinal waves in a cylindrical Kirchhoff–Love shell interacting with a nonlinear elastic medium is investigated. A generalized model of a nonlinear elastic medium is introduced into consideration, which is reduced in particular cases to the well-known two-parameter Pasternak and Hetenyi models. Using the asymptotic multiscale method for the component of the longitudinal displacement, a non-integrable quasi-hyperbolic equation of the sixth order is derived, generalizing the Ostrovsky equation, and classes of its exact solutions are constructed in terms of the Jacobi elliptic function. It is demonstrated that exact periodic solutions are physically realizable, while solitary wave solutions are nonphysical. The further asymptotic analysis of the derived equation made it possible to obtain the nonlinear Schrödinger equation and use the Lighthill criterion. Relationships are established between the parameters of a nonlinear elastic medium, which are necessary for the development of the modulation instability of small periodic waves.

NUMERICAL INVESTIGATION OF THE NONLINEAR FRACTIONAL OSTROVSKY EQUATION

This research paper investigates the numerical solutions of the nonlinear fractional Ostrovsky equation through five recent numerical schemes (Adomian decomposition (AD), El Kalla (EK), Cubic B-Spline (CBS), extended Cubic B-Spline (ECBS), exponential Cubic B-Spline (ExCBS) schemes). We investigate the obtained computational solutions via the generalized Jacobi elliptical functional (JEF) and modified Khater (MK) methods. This model is considered as a mathematical modification model of the Korteweg–de Vries (KdV) equation with respect to the effects of background rotation. The solitary solutions of the well-known mathematical model (KdV equation) usually decay and are replaced by radiating inertia gravity waves. The obtained solitary solutions show the localized wave packet as a persistent and dominant feature. The accuracy of the obtained numerical solutions is investigated by calculating the absolute error between the exact and numerical solutions. Many sketches are given to illustrate the matching between the exact and numerical solutions.

Symmetries, conservation laws, and generalized travelling waves for a forced Ostrovsky equation

Ostrovsky’s equation with time- and space-dependent forcing is studied. This equation is model for long waves in a rotating fluid with a non-constant depth (topography). A classification of Lie point symmetries and low-order conservation laws is presented. Generalized travelling wave solutions are obtained through symmetry reduction. These solutions exhibit a wave profile that is stationary in a moving reference frame whose speed can be constant, accelerating, or decelerating.

The Cauchy problem for the generalized Ostrovsky equation with negative dispersion

This paper is devoted to studying the Cauchy problem for the generalized Ostrovsky equation \begin{eqnarray*} u_{t}-\beta\partial_{x}^{3}u-\gamma\partial_{x}^{-1}u+\frac{1}{k+1}(u^{k+1})_{x}=0,k\geq5 \end{eqnarray*} with $\beta\gamma<0,\gamma>0$. Firstly, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $H^{s}(\mathbb{R})\left(s>\frac{1}{2}-\frac{2}{k}\right)$. Then, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $X_{s}(\mathbb{R}): =\|f\|_{H^{s}}+\left\|\mathscr{F}_{x}^{-1}\left(\frac{\mathscr{F}_{x} f(\xi)}{\xi}\right)\right\|_{H^{s}}\left(s>\frac{1}{2}-\frac{2}{k}\right).$ Finally, we show that the solution to the Cauchy problem for generalized Ostrovsky equation converges to the solution to the generalized KdV equation as the rotation parameter $\gamma$ tends to zero for data belonging to $X_{s}(\mathbb{R})(s>\frac{3}{2})$. The main difficulty is that the phase function of Ostrosvky equation with negative dispersive $\beta\xi^{3}+\frac{\gamma}{\xi}$ possesses the zero singular point.

On the Dimension of the Divergence Set of the Ostrovsky Equation

On the Dimension of the Divergence Set of the Ostrovsky Equation

New abundant solitary wave structures for a variety of some nonlinear models of surface wave propagation with their geometric interpretations

The propagation of waves of water on the surface is characterized by various mathematical models. In this work, we adopt the simplest equation method as well as the Kudryashov's new function method, to a variety of some nonlinear models of surface wave propagation to extract their abundant solitary wave structures. These nonlinear models include the Benjamin‐Bona‐Mahony (BBM) equation, the Ostrovsky (OS) equation, the Ostrovsky‐Benjamin‐Bona‐Mahony (OS‐BBM) equation, and the Boussinesq system of equations. Based on the solutions of two distinct auxiliary equations, various wave solutions are successfully obtained. In addition, the geometric interpretations of the obtained solutions are analyzed. To illuminate the physical nature of the obtained solutions of this paper, some graphical representations (numerical simulations) of the acquired solutions have been depicted in two‐dimensional density, two‐ and three‐dimensional graphs. Furthermore, a comparison between the obtained results of this paper and the solutions for the considered models obtained in the literature is also provided. It is shown that the used approaches are very effective and powerful strategies for a wider class of nonlinear partial differential equations in physical problems.

High-Frequency Instabilities of the Ostrovsky Equation

We study the spectral stability of small-amplitude periodic traveling waves of the Ostrovsky equation. We prove that these waves exhibit spectral instabilities arising from a collision of pair of non-zero eigenvalues on the imaginary axis when subjected to square-integrable perturbations on the whole real line. We also list all such collisions between pairs of eigenvalues on the imaginary axis and do a Krein signature analysis.

Hamiltonian description of internal ocean waves with Coriolis force

&lt;p style="text-indent:20px;"&gt;The interfacial internal waves are formed at the pycnocline or thermocline in the ocean and are influenced by the Coriolis force due to the Earth's rotation. A derivation of the model equations for the internal wave propagation taking into account the Coriolis effect is proposed. It is based on the Hamiltonian formulation of the internal wave dynamics in the irrotational case, appropriately extended to a &lt;em&gt;nearly&lt;/em&gt; Hamiltonian formulation which incorporates the Coriolis forces. Two propagation regimes are examined, the long-wave and the intermediate long-wave propagation with a small amplitude approximation for certain geophysical scales of the physical variables. The obtained models are of the type of the well-known Ostrovsky equation and describe the wave propagation over the two spatial horizontal dimensions of the ocean surface.&lt;/p&gt;

Mathematical analysis of a conservative numerical scheme for the Ostrovsky equation

Mathematical analysis of a conservative numerical scheme for the Ostrovsky equation