Beyond the KdV: Post-explosion development

Abstract

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.

Dynamics of Internal Envelope Solitons in a Rotating Fluid of a Variable Depth

“…The effect of Earth' rotation on the dynamics of nonlinear waves in the oceans has been extensively studied in previous decades (see, for example, Refs. [24,11,14,12,20,18,25,31] and references therein). As is well-know, wave propagation in big lakes can be also affected by the Earth's rotation [5,7,28,32,29].…”

Formation of wave packets in the Ostrovsky equation for both normal and anomalous dispersion

“…The Ostrovsky equation (u t + αuu x + βu xxx ) x = γ u (1.1) describes weakly nonlinear waves in continuous media with two kinds of dispersion, the smallscale dispersion typical for the Koretweg-de Vries (KdV) equation and large-scale dispersion typical for electromagnetic or optic waves in waveguides, or due to the effect of background rotation for internal and surface waves [1][2][3][4][5].…”

The effects of interplay between the rotation and shoaling for a solitary wave on variable topography

“…The Ostrovsky model is formally inapplicable to such strong waves, but fortunately, it still provides quite reasonable estimates even in such cases (see, eg, Refs. 22,28,29).…”

“…For such a hydrology configuration when h 2 < h 1 , the bottom profile providing soliton propagation in a rotating fluid with the constant amplitude is also possible. It follows from the solution of Equation 29 where h 2 and h 1 should be inter-replaced; the corresponding solution reduces again to Equation 31 and is shown in Figure 11 by line 3.…”