The Reduced Ostrovsky Equation: Integrability and Breaking

Grimshaw, Roger; Helfrich, Karl R.; Johnson, E. R.

doi:10.1111/j.1467-9590.2012.00560.x

Cited by 44 publications

(66 citation statements)

References 26 publications

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“…It has been shown in Ref. 38 that if the initial condition u(x, 0) ¼ u 0 (x) is such that d 2 u 0 (x)/dx 2 > c/ 3a at some x, then wave breaking eventually occurs and the solution becomes singular. If d 2 u 0 (x)/dx 2 < c/3a for all x, the solution remains smooth at all times.…”

Section: B Reduced Rkdvmentioning

confidence: 99%

Beyond the KdV: Post-explosion development

Ostrovsky

Pelinovsky

Shrira

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Several threads of the last 25 years' developments in nonlinear wave theory that stem from the classical Korteweg-de Vries (KdV) equation are surveyed. The focus is on various generalizations of the KdV equation which include higher-order nonlinearity, large-scale dispersion, and a nonlocal integral dispersion. We also discuss how relatively simple models can capture strongly nonlinear dynamics and how various modifications of the KdV equation lead to qualitatively new, non-trivial solutions and regimes of evolution observable in the laboratory and in nature. As the main physical example, we choose internal gravity waves in the ocean for which all these models are applicable and have genuine importance. We also briefly outline the authors' view of the future development of the chosen lines of nonlinear wave theory. V C 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4927448]In the aftermath of the revolutionary development in the theory of nonlinear waves in the 1960s-1970s, the intensity of studies in this field does not show signs of decreasing. Here, several lines of studies related to the KdV equation are surveyed. Focusing on generalizations of the KdV equation, we trace the development of main ideas and concepts of nonlinear wave theory yielding qualitatively new solutions such as "fat" and table-top solitons, breathers, and slowly radiating solitons. The reasons underpinning the unmatched universality of the KdV equation as a mathematical model applicable in many physical contexts are quite natural: for long waves the model combines the most typical, small quadratic nonlinearity and weak, small-scale dispersion. The balance between the nonlinearity and dispersion allows the possibility of solitary waves possessing astonishing particlelike properties such as robustness and persistence in collisions not only with each other but also with other perturbations. This particle-like behavior is at the origin of the term soliton introduced to emphasize the affinity of such waves with elementary particles (electrons, protons, etc). As the understanding of nonlinear waves matured, the limitations of the KdV model and necessity to go beyond it became apparent; hence, the trend towards developing more general and rich models which generalize the KdV equation and yield many new and non-trivial results. Here, we briefly discuss their appearance in various mathematical and physical contexts and some of the results which follow, such as qualitatively new types of solitons, their limiting shapes and parameters, and interactions. It is also demonstrated that these features are not mathematical artefacts, which is illustrated by examples mainly related to nonlinear internal gravity waves in the ocean.

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Section: B Reduced Rkdvmentioning

confidence: 99%

Beyond the KdV: Post-explosion development

Ostrovsky

Pelinovsky

Shrira

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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show abstract

“…Once the wave front steepens sufficiently it disperses into a primary wave packet and a tail of smaller dispersive waves. We demonstrated that the nonlinear wave packet interpretation of the wave train of Grimshaw et al (2012) is appropriate in some parameter regimes, with changes in amplitude reflected in the phase of the nearly solitary wave response. By mapping out the parameter space we have shown that, as expected, the Rossby number is the controlling variable for the dynamics in this problem.…”

Section: Discussionmentioning

confidence: 99%

“…The KdV equation is the simplest model equation that allows for a balance between nonlinear and dispersive effects, with a rich mathematical structure which makes predictions of the evolution of an initial state that are remarkably robust in both laboratory and field settings (see Johnson, 1997). A rotation-modified version of the KdV equation was first derived by Ostrovsky (see Grimshaw et al, 2012 for an in-depth discussion of the equation properties and references to the Russian literature). This new equation was subsequently analysed both through theoretical solutions found by asymptotic expansions and through numerical solutions.…”

Section: Introductionmentioning

confidence: 99%

The fully nonlinear stratified geostrophic adjustment problem

Coutino

Stastna

2017

Nonlin. Processes Geophys.

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Abstract. The study of the adjustment to equilibrium by a stratified fluid in a rotating reference frame is a classical problem in geophysical fluid dynamics. We consider the fully nonlinear, stratified adjustment problem from a numerical point of view. We present results of smoothed dam break simulations based on experiments in the published literature, with a focus on both the wave trains that propagate away from the nascent geostrophic state and the geostrophic state itself. We demonstrate that for Rossby numbers in excess of roughly 2 the wave train cannot be interpreted in terms of linear theory. This wave train consists of a leading solitarylike packet and a trailing tail of dispersive waves. However, it is found that the leading wave packet never completely separates from the trailing tail. Somewhat surprisingly, the inertial oscillations associated with the geostrophic state exhibit evidence of nonlinearity even when the Rossby number falls below 1. We vary the width of the initial disturbance and the rotation rate so as to keep the Rossby number fixed, and find that while the qualitative response remains consistent, the Froude number varies, and these variations are manifested in the form of the emanating wave train. For wider initial disturbances we find clear evidence of a wave train that initially propagates toward the near wall, reflects, and propagates away from the geostrophic state behind the leading wave train. We compare kinetic energy inside and outside of the geostrophic state, finding that for long times a Rossby number of around one-quarter yields an equal split between the two, with lower (higher) Rossby numbers yielding more energy in the geostrophic state (wave train). Finally we compare the energetics of the geostrophic state as the Rossby number varies, finding long-lived inertial oscillations in the majority of the cases and a general agreement with the past literature that employed either hydrostatic, shallow-water equation-based theory or stratified Navier-Stokes equations with a linear stratification.

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“…This construction goes back to at least the late 70's and it was revisited in several publications, [19,5,23,1,6]. In fact, the authors in [6] placed special emphasis of these explicit solutions and asked about further properties of these simple solutions.…”

Section: Spectral Stability For the Periodic Waves In The Quadratic Casementioning

confidence: 99%

“…Clearly, (6), being a fully nonlinear equation, is not a very nice object to deal with. Thus, we perform a (solution dependent) change of variables, namely…”

Section: Introductionmentioning

confidence: 99%

Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models

2017

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Abstract. We consider the Ostrovsky and short pulse models in a symmetric spatial interval, subject to periodic boundary conditions. For the Ostrovsky case, we revisit the classical periodic traveling waves and for the short pulse model, we explicitly construct traveling waves in terms of Jacobi elliptic functions. For both examples, we show spectral stability, for all values of the parameters. This is achieved by studying the non-standard eigenvalue problems in the form Lu = λu , where L is a Hill operator.

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The Reduced Ostrovsky Equation: Integrability and Breaking

Cited by 44 publications

References 26 publications

Beyond the KdV: Post-explosion development

Beyond the KdV: Post-explosion development

The fully nonlinear stratified geostrophic adjustment problem

Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models

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