Ostrovsky Equation Research Articles

Nonlinear waves in magnetized quark matter and the reduced Ostrovsky equation

We study nonlinear waves in a nonrelativistic ideal and cold quark gluon plasma immersed in a strong uniform magnetic field. In the context of nonrelativistic hydrodynamics with an external magnetic field we derive a nonlinear wave equation for baryon density perturbations, which can be written as a reduced Ostrovsky equation. We find analytical solutions and identify the effects of the magnetic field.

Well-posedness result for the Ostrovsky, Stepanyams and Tsimring equation at the critical regularity

Studied here is the Ostrovsky, Stepanyams and Tsimring (OST) equation ut+uxxx−εH(u+uxx)x+uux=0. It is showed that the associated initial value problem is locally analytically well-posed in the critical Sobolev spaces H−3∕2(R). This improves the previous results on this equation in the Sobolev spaces.

Airy functions and Riesz fractional trigonometric operators

ABSTRACTIn this paper, using the Riesz fractional derivative , we present the Mellin transforms of shifted Airy functions in terms of the Riesz fractional derivatives of Airy functions. We also show some integral addition formulas for the products of Airy functions in terms of the Riesz fractional trigonometric operators. Moreover, the fundamental solution of the Ostrovsky equation is given by the Riesz fractional trigonometric operators.

The Cauchy problem for the Ostrovsky equation with positive dispersion

This paper is devoted to studying the Cauchy problem for the Ostrovsky equation $$\begin{aligned} \partial _{x}\left( u_{t}-\beta \partial _{x}^{3}u +\frac{1}{2}\partial _{x}(u^{2})\right) -\gamma u=0, \end{aligned}$$with positive \(\beta \) and \(\gamma \). This equation describes the propagation of surface waves in a rotating oceanic flow. We first prove that the problem is locally well-posed in \(H^{-\frac{3}{4}}(\text{ R })\). Then we reestablish the bilinear estimate, by means of the Strichartz estimates instead of calculus inequalities and Cauchy–Schwartz inequalities. As a byproduct, this bilinear estimate leads to the proof of the local well-posedness of the problem in \(H^{s}(\text{ R })\) for \( s>-\frac{3}{4}\), with help of a fixed point argument.

The Cauchy problem for quadratic and cubic Ostrovsky equation with negative dispersion

In this paper, we consider the Cauchy problem for the quadratic and cubic Ostrovsky equation with negative dispersion ∂xut−β∂x3u+1k∂x(uk)−γu=0,β<0,γ>0,(k=2,3).Firstly, by using the Strichartz estimates instead of the Cauchy–Schwarz inequalities, we give an alternative proof of Lemma 1.2 of Isaza and Mejía (2006). Secondly, by using the Strichartz estimates instead of the Cauchy–Schwarz inequalities, we give an alternative proof of Lemma 1.3 of Isaza and Mejía (2007). Thirdly, we prove that the Cauchy problem for the cubic Ostrovsky equation is locally well-posed in Hs(R) with s≥14. Finally, we prove that the Cauchy problem for the cubic Ostrovsky equation is not well-posed in Hs(R) with s<14.

The Evolution of Internal Undular Bores over a Slope in the Presence of Rotation

AbstractThe 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.

Adiabatic decay of internal solitons due to Earth's rotation within the framework of the Gardner-Ostrovsky equation.

The adiabatic decay of different types of internal wave solitons caused by the Earth's 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: (ut + c ux + α u ux + α1 u2 ux + β uxxx)x = γ u. It is shown that at the early stage of evolution solitons gradually decay under the influence of weak Earth's rotation described by the parameter γ. The characteristic decay time is derived for different types of solitons for positive and negative coefficients of cubic nonlinearity α1 (both signs of that parameter may occur in the oceans). The coefficient of quadratic nonlinearity α determines only a polarity of solitary wave when α1 < 0 or the asymmetry of solitary waves of opposite polarity when α1 > 0. 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 α1 is negative, are structurally unstable. Under the influence of Earth's 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.

Traveling wave solutions for a generalized Ostrovsky equation

AbstractIn this paper looking for traveling wave solutions, we find that when the polynomial of velocity is quintic the generalized Ostrovsky equation (GOE) has abundant exact solutions that can be expressed in terms of the Jacobi elliptic functions. Hence, the GOE has a plenty of periodic waves, solitary waves, compactons, etc. These solutions are derived from the solutions of a simple non‐linear ordinary differential equation. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Nonlinear waves in an elastic guide interacting with an elastic-inertial foundation

This paper deals with the study of the transverse wave propagation in a string resting on an elastic-inertial foundation. A self-consistent dynamic problem is herein considered for a system composed of a one-dimensional flexible guide (string) and an elastic-inertial foundation. As a foundation model the Vesnitsky’s model was chosen. The set of equations is reduced to one fourth-order cubic-nonlinear equation relative to transverse displacements of the string. Depending on the string and the elastic-inertial foundation mass ratio, the evolutionary equation has the following three extreme events: the modified Ostrovsky equation, the Riemann equation with a cubic nonlinearity, the equation for an anharmonic oscillator with a cubic nonlinearity.

Investigation of Dark and Bright Soliton Solutions of Some Nonlinear Evolution Equations

In this paper, generalized Kudryashov method (GKM) is used to find the exact solutions of (1+1) dimensional nonlinear Ostrovsky equation and (4+1) dimensional Fokas equation. Firstly, we get dark and bright soliton solutions of these equations using GKM. Then, we remark the results we found using this method.

On the Invariant Measure for the Ostrovsky Equation

In this paper, we construct invariant measure for the Ostrovsky equation associated with conservation laws. On the other hand, we prove the local well-posedness of the initial value problem for the periodic Ostrovsky equation with initial data in $\phantom {\dot {i}}H^{s}(\mathbb {T})$ for $\phantom {\dot {i}}s\geq -\frac {1}{2}$ .

A note on the Ostrovsky equation in weighted Sobolev spaces

In this work we consider the initial value problem (IVP) associated to the Ostrovsky equationsut+∂x3u±∂x−1u+u∂xu=0,x∈R,t∈R,u(x,0)=u0(x).} We study the well-posedness of the IVP in the weighted Sobolev spacesZ˙s,s2:={f∈Hs(R):∂x−1f∈L2(R)}∩L2(|x|sdx), with 34<s≤1.

Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion

The goal of this paper is two-fold. Firstly, by using the Fourier restriction norm method and the fixed point theorem, we prove that the Cauchy problem for a generalized Ostrovsky equation \t\t\t∂x(ut−β∂x3u+13∂x(u3))−γu=0,β>0,γ>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\partial_{x} \\biggl(u_{t}-\\beta\\partial_{x}^{3}u +\\frac{1}{3}\\partial_{x}\\bigl(u^{3}\\bigr) \\biggr) - \\gamma u=0,\\quad\\beta>0,\\gamma>0, $$\\end{document} is locally well-posed in H^{s}(mathbf{R}) with sgeqfrac{1}{4}. Secondly, we prove that the Cauchy problem for a generalized Ostrovsky equation is not well-posed in H^{s}(mathbf{R}) with s<frac{1}{4} in the sense that the solution map is C^{3}.

Well-posedness and unique continuation property for the solutions to the generalized Kawahara equation below the energy space

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Kawahara equation [Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation, Adv Math (China).45(2016),pp.80–88] as follows: with initial data in the Sobolev space Benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488–2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ; Acta Math Sci.37(2017),pp.385C39], first, we show that the local well-posedness is established for the initial data with () and () respectively. Then,using these results and conservation laws, we also prove that the IVP is globally well-posed for the initial data with (). Finally, benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488–2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation,Adv Math (China).45(2016),pp.80–88], i.e. using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property (UCP henceforth) for the IVP.

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

The large-amplitude internal solitary waves commonly observed in the coastal ocean can propagate for long distances for long times, so that it may be necessary to take account of the effects of the Earth’s background rotation. In this case an appropriate model wave evolution equation is the Ostrovsky equation, whose typical solutions indicate that internal solitary waves will evolve into envelope wave packets. Unlike the more usual Korteweg-de Vries solutions which are typically rank-ordered wave packets, these are centred with the largest waves in the middle. This qualitative feature, together with certain key quantitative parameters such as the envelope carrier wavenumber and speed, can be sought in oceanic observations. Hence we have examined many SAR images of internal solitary waves with the general aim of finding features indicating that rotational effects have become significant. From these we report in detail on six typical cases of which four give indications of rotational effects. In addition we use a two-layer fluid model to estimate how the rotational parameters depend on the background stratification and topography.

Spectral stability of periodic waves in the generalized reduced Ostrovsky equation

We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent literature, we give a simple argument that proves spectral stability of all smooth periodic travelling waves independent of the nonlinearity power. The argument is based on the energy convexity and does not use coordinate transformations of the reduced Ostrovsky equations to the semi-linear equations of the Klein–Gordon type.

Nonlinear propagation of vector extremely short pulses in a medium of symmetric and asymmetric molecules

The nonlinear propagation of extremely short electromagnetic pulses in a medium of symmetric and asymmetric molecules placed in static magnetic and electric fields is theoretically studied. Asymmetric molecules differ in that they have nonzero permanent dipole moments in stationary quantum states. A system of wave equations is derived for the ordinary and extraordinary components of pulses. It is shown that this system can be reduced in some cases to a system of coupled Ostrovsky equations and to the equation intagrable by the method for an inverse scattering transformation, including the vector version of the Ostrovsky–Vakhnenko equation. Different types of solutions of this system are considered. Only solutions representing the superposition of periodic solutions are single-valued, whereas soliton and breather solutions are multivalued.

A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue

In the present paper, we propose a two-component generalization of the reduced Ostrovsky (Vakhnenko) equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis–Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that the two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and N-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Bäcklund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of the two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. In particular, the backward difference form of above semi-discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their N-soliton solutions in terms of pffafians are also provided.

Whitham modulation theory for the Ostrovsky equation.

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.

Periodic Traveling Waves of the Regularized Short Pulse and Ostrovsky Equations: Existence and Stability

We construct various periodic traveling wave solutions of the Ostrovsky/Hunter--Saxton/short pulse equation and its KdV regularized version. For the regularized short pulse model with small Coriolis parameter, we describe a family of periodic traveling waves which are a perturbation of appropriate KdV solitary waves. We show that these waves are spectrally stable. For the short pulse model, we construct a family of traveling peakons with corner crests. We show that the peakons are spectrally stable as well.