Nonlinear Surface and Internal Waves in Rotating Fluids

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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“…35. It was found that at certain conditions there exist nonlinear periodic waves consisting of parabolic sections mentioned above for Eq.…”

Section: Soliton Evolution In Rkdvmentioning

confidence: 97%

Beyond the KdV: Post-explosion development

Ostrovsky

Pelinovsky

Shrira

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…We omit further index n and present Eq. u(x, t) dx = 0 (see, for instance, [9,15,19] where this property is explained for the Ostrovsky equation which is a particular case of the GO equation (1.1) with α 1 = 0). To approximate partial derivatives in Eq.…”

Section: Numerical Scheme For the Solution Of The Gardner-ostrovsky Ementioning

confidence: 99%

“…The Ostrovsky equation originally derived for the description of weakly nonlinear waves in a rotating ocean [14] can be treated as the particular case of the GO equation when the cubic nonlinear term in Eq. (1.1) is omitted (α 1 = 0).…”

Section: Terminal Decay Of a Kdv Soliton Due To Influence Of A Large-mentioning

confidence: 99%

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On Numerical Solution of the Gardner–Ostrovsky Equation

Obregon

Stepanyants

2012

Math. Model. Nat. Phenom.

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Abstract. A simple explicit numerical scheme is proposed for the solution of the GardnerOstrovsky equation ut + cux + αuux + α1u 2 ux + βuxxx x = γu which is also known as the extended rotation-modified Korteweg-de Vries (KdV) equation. This equation is used for the description of internal oceanic waves affected by Earth' rotation. Particular versions of this equation with zero some of coefficients, α, α1, β, or γ are also known in numerous applications. The proposed numerical scheme is a further development of the well-known finite-difference scheme earlier used for the solution of the KdV equation. The scheme is of the second order accuracy both on temporal and spatial variables. The stability analysis of the scheme is presented for infinitesimal perturbations. The conditions for the calculations with the appropriate accuracy have been found. Examples of calculations with the periodic boundary conditions are presented to illustrate the robustness of the proposed scheme.

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“…It is often called the OstrovskyVakhnenko (OV) Eq. [11] and was initially constructed for other physical systems, like for internal waves in rotating ocean [12], and for finding solutions to mathematical physics models similar to solitons [13]. An analysis of an even more general OV equation with varying coefficients was done in Ref.…”

Section: Introductionmentioning

confidence: 99%

A new equation and exact solutions describing focal fields in media with modular nonlinearity

Руденко

Hedberg

2017

Nonlinear Dyn

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Brand-new equations which can be regarded as modifications of Khokhlov-ZabolotskayaKuznetsov or Ostrovsky-Vakhnenko equations are suggested. These equations are quite general in that they describe the nonlinear wave dynamics in media with modular nonlinearity. Such media exist among composites, meta-materials, inhomogeneous and multiphase systems. These new models are interesting because of two reasons: (1) the equations admit exact analytic solutions and (2)

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Nonlinear Surface and Internal Waves in Rotating Fluids

Cited by 193 publications

References 5 publications

Beyond the KdV: Post-explosion development

Beyond the KdV: Post-explosion development

On Numerical Solution of the Gardner–Ostrovsky Equation

A new equation and exact solutions describing focal fields in media with modular nonlinearity

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