Ostrovsky Equation Research Articles

On the generalized reduced Ostrovsky equation

We survey recent progress on the case of the Cauchy problem for the generalized reduced Ostrovsky equation ut=S(∂x)u+(f(u))x, where the operator S(∂x) is defined through the Fourier transform as S(∂x)=ℱ−11iξℱ, and the nonlinear interaction is given by f(u)=|u|ρ−1u if ρ> 1 is not an integer and f(u)=uρ if ρ> 1 is an integer.

High accuracy numerical methods for the Gardner–Ostrovsky equation

The Gardner–Ostrovsky equation, also known as the extended rotation-modified Korteweg–de Vries (KdV) equation, describes weakly nonlinear internal oceanic waves under the influence of Earth’ rotation. High accuracy numerical methods are needed to follow with precision the long time evolution of the solutions of this equation, with the additional difficulty that the numerical methods have to conserve very accurately several invariants of the solutions, including the mass and the energy of the waves. Finite-difference methods traditionally used for the solution of the KdV equation fails to preserve accurately these invariants for large times. In this paper we show that this difficulty can be overcome by using a high accuracy finite-difference (HAFD) numerical method. We present a strong-stability-preserving finite-difference scheme and compare its performance, using some relevant examples, with those of two recently published numerical methods for solving this kind of equation: a simpler second order finite-difference method and a pseudospectral numerical scheme that enforces the conservation of energy. The numerical comparison shows that the three methods have similar accuracy for short times, but the simpler finite-difference scheme is less accurate in preserving the three invariants, affecting the numerical accuracy of the solution as time goes on. On the contrary, the HAFD method presented here preserves the invariants with even better accuracy than the pseudospectral scheme, but with a much lower computational cost. In addition, the numerical implementation of the HAFD method is as easy as that of the simpler finite-difference method, being both much simpler than the intricate energy conservation pseudospectral scheme. These advantages makes the HAFD method presented here very appropriate for solving numerically this type of equations, particularly for studying long time wave propagation.

Combined Effect of Rotation and Topography on Shoaling Oceanic Internal Solitary Waves

Abstract Internal solitary waves commonly observed in the coastal ocean are often modeled by a nonlinear evolution equation of the Korteweg–de Vries type. Because these waves often propagate for long distances over several inertial periods, the effect of Earth’s background rotation is potentially significant. The relevant extension of the Kortweg–de Vries is then the Ostrovsky equation, which for internal waves does not support a steady solitary wave solution. Recent studies using a combination of asymptotic theory, numerical simulations, and laboratory experiments have shown that the long time effect of rotation is the destruction of the initial internal solitary wave by the radiation of small-amplitude inertia–gravity waves, and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. However, in the ocean, internal solitary waves are often propagating over variable topography, and this alone can cause quite dramatic deformation and transformation of an internal solitary wave. Hence, the combined effects of background rotation and variable topography are examined. Then the Ostrovsky equation is replaced by a variable coefficient Ostrovsky equation whose coefficients depend explicitly on the spatial coordinate. Some numerical simulations of this equation, together with analogous simulations using the Massachusetts Institute of Technology General Circulation Model (MITgcm), for a certain cross section of the South China Sea are presented. These demonstrate that the combined effect of shoaling and rotation is to induce a secondary trailing wave packet, induced by enhanced radiation from the leading wave.

Validity of the Weakly Nonlinear Solution of the Cauchy Problem for the Boussinesq‐Type Equation

We consider the initial‐value problem for the regularized Boussinesq‐type equation in the class of periodic functions. Validity of the weakly nonlinear solution, given in terms of two counterpropagating waves satisfying the uncoupled Ostrovsky equations, is examined. We prove analytically and illustrate numerically that the improved accuracy of the solution can be achieved at the timescales of the Ostrovsky equation if solutions of the linearized Ostrovsky equations are incorporated into the asymptotic solution. Compared to the previous literature, we show that the approximation error can be controlled in the energy space of periodic functions and the nonzero mean values of the periodic functions can be naturally incorporated in the justification analysis.

Convergence of the Ostrovsky equation to the Ostrovsky–Hunter one

We consider the Ostrovsky equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the Ostrovsky–Hunter equation. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the Lp setting.

New Applications to Solitary Wave Ansatz

In this article, the solitary wave and shock wave solitons for nonlinear Ostrovsky equation and Potential Kadomstev-Petviashvili equations have been obtained. The solitary wave ansatz is used to carry out the solutions.

Coupled Ostrovsky Equations for Internal Waves, with a Background Shear Flow

We study the behaviour of weakly nonlinear oceanic internal waves in the presence of background rotation and shear flow, when two distinct linear long wave modes have nearly coincident phase speeds. The waves are described by a system of coupled Ostrovsky equations, derived from the full set of Euler equations for incompressible density stratified fluid with a free surface and rigid bottom boundary conditions. We report here some preliminary results obtained when there is a background shear flow.

Modified reduced Ostrovsky equation: Integrability and breaking

The modified reduced Ostrovsky equation is a reduction of the modified Korteweg-de Vries equation, in which the usual linear dispersive term with a third-order derivative is replaced by a linear nonlocal integral term, representing the effect of background rotation. Here we study the case when the cubic nonlinear term has the same polarity as the rotation term. This equation is integrable provided certain slope constraints are satisfied. We demonstrate, through theoretical analysis and numerical simulations, that when this constraint is not satisfied at the initial time, wave breaking inevitably occurs.

Global existence of small-norm solutions in the reduced Ostrovsky equation

We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitzeica equation and prove global existence of small-norm solutions in Sobolev space $H^3(\mathbb{R})$. This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reduced Ostrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking.

Asymptotics of solutions to the generalized Ostrovsky equation

We consider the Cauchy problem for the generalized Ostrovsky equationutx=u+(f(u))xx, where f(u)=|u|ρ−1u if ρ is not an integer and f(u)=uρ if ρ is an integer. We obtain the L∞ time decay estimates and the large time asymptotics of small solutions under suitable conditions on the initial data and the order of the nonlinearity.

On strongly interacting internal waves in a rotating ocean and coupled Ostrovsky equations

In the weakly nonlinear limit, oceanic internal solitary waves for a single linear long wave mode are described by the KdV equation, extended to the Ostrovsky equation in the presence of background rotation. In this paper we consider the scenario when two different linear long wave modes have nearly coincident phase speeds and show that the appropriate model is a system of two coupled Ostrovsky equations. These are systematically derived for a density-stratified ocean. Some preliminary numerical simulations are reported which show that, in the generic case, initial solitary-like waves are destroyed and replaced by two coupled nonlinear wave packets, being the counterpart of the same phenomenon in the single Ostrovsky equation.

Unique continuation principle for the Ostrovsky equation with negative dispersion

In this article we prove that if the difference of two solutions of the Ostrovsky equation with negative dispersion,∂tu+∂x3u−∂xu+u∂xu=0, has certain exponential decay for x>0 at two different times, then both solutions are equal.

Random attractors of the stochastic damped forced Ostrovsky equation

The existence of a random attractor is established for the stochastic damped forced Ostrovsky equation with additive noise in L̃2(R). The main method is the combination of the energy equation, a suitable splitting of the solution, and the bilinear estimates in Bourgain spaces.

Analytical and Multishaped Solitary Wave Solutions for Extended Reduced Ostrovsky Equation

We present the analytical and multishaped solitary wave solutions for extended reduced Ostrovsky equation (EX-ROE). The exact solitary (traveling) wave solutions are expressed by three types of functions which are hyperbolic function solution, trigonometric function solution, and rational solution. These results generalized the previous results. Multishape solitary wave solutions such as loop-shaped, cusp-shaped, and hump-shaped can be obtained as well when the special values of the parameters are taken. The -expansion method presents a wide applicability for handling nonlinear partial differential equations.

On the stability of solitary waves of a generalized Ostrovsky equation

We consider the stability of ground state solitary waves of the generalized Ostrovsky equation $$( u_t - \beta u_{xxx} + f(u)_x)_x = \gamma u$$ , with homogeneous nonlinearities of the form $$f(u)=a_e|u|^p+a_o|u|^{p-1}u$$ . We obtain bounds on the function $$d$$ whose convexity determines the stability of the solitary waves. These bounds imply that, when $$2\le p<5$$ and $$a_o<0$$ , solitary waves are stable for $$c$$ near $$c_*=2\sqrt{\beta \gamma }$$ . These bounds also imply that, for $$\gamma >0$$ small, solitary waves are stable when $$2\le p<5$$ and unstable when $$p>5$$ . We also numerically compute the function $$d$$ , and thereby determine precise regions of stability and instability, for several nonlinearities.

The Reduced Ostrovsky Equation: Integrability and Breaking

The reduced Ostrovsky equation is a modification of the Korteweg‐de Vries equation, in which the usual linear dispersive term with a third‐order derivative is replaced by a linear nonlocal integral term, which represents the effect of background rotation. This equation is integrable provided a certain curvature constraint is satisfied. We demonstrate, through theoretical analysis and numerical simulations, that when this curvature constraint is not satisfied at the initial time, then wave breaking inevitably occurs.

On the τ-functions of the reduced Ostrovsky equation and the A(2)2 two-dimensional Toda system

The reciprocal link between the reduced Ostrovsky equation and the A(2)2 two-dimensional Toda (2D-Toda) system is used to construct the N-soliton solution of the reduced Ostrovsky equation. The N-soliton solution of the reduced Ostrovsky equation is presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations and the τ-function of the reduced Ostrovsky equation are obtained from the period 3-reduction of the B∞ or C∞ 2D-Toda system, i.e. the A(2)2 2D-Toda system. One of the τ-functions of the A(2)2 2D-Toda system becomes the square of a pfaffian which also becomes a solution of the reduced Ostrovsky equation. There is another bilinear equation which is a member of the 3-reduced extended BKP hierarchy. Using this bilinear equation, we can also construct the same pfaffian solution.

Evolution Solutions for the Ostrovsky Equation: A Perspective from an Auxiliary Elliptic Equation

The Ostrovsky equation is used to describe the oceanic waves affected by Earth rotation. The auxiliary elliptic equation was employed as a direct method to construct the evolution solutions for the Ostrovsky equation in this manuscript. Detailed analysis of evolution solutions of the Ostrovsky equation is presented. The periodic solutions, which read Jacobi elliptic function solutions, hyperbolic solutions and trigonal solutions, are obtained. More parameters are included in the evolution solutions, the parameter space is enlarged. This method adds a new route to explore evolution solutions of nonlinear differential equation in a perspective from auxiliary elliptic equation.

The effect of rotation on internal solitary waves

In the weakly nonlinear long-wave regime, internal solitary waves are often modelled by the Korteweg– de Vries equation, which is well known to support an exact solitary wave solution. However, when the effect of background rotation is taken into account, an additional term is needed and the outcome is the Ostrovsky equation. Although the additional term would appear to be relatively mild, being a linear long-wave perturbation, it has the drastic effect of destroying the solitary wave solution. Instead an initial solitary-like disturbance decays into radiating oscillatory waves, with the eventual formation of a nonlinear envelope wave packet, whose carrier wavenumber is determined by an extremum in the group velocity. In this paper, we will use a combination of theoretical analyses, numerical simulations and laboratory experiments to describe this process.

On uniqueness properties of solutions of the Ostrovsky equation with positive dispersion

In this article we prove that if the difference of two sufficiently smooth solutions of the Ostrovsky equation with positive dispersion ∂tu+∂x3u+∂x−1u+u∂xu=0, has certain exponential decay for x>0, at two different times, then they are equal.