2010
DOI: 10.1137/09075799x
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Wave Breaking in the Ostrovsky–Hunter Equation

Abstract: The Ostrovsky-Hunter equation governs evolution of shallow water waves on a rotating fluid in the limit of small high-frequency dispersion. Sufficient conditions for the wave breaking in the Ostrovsky-Hunter equation are found both on an infinite line and in a periodic domain. Using the method of characteristics, we also specify the blow-up rate at which the waves break. Numerical illustrations of the finite-time wave breaking are given in a periodic domain.

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Cited by 52 publications
(45 citation statements)
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“…However, the full equation is not integrable, with a breakdown on the lines X = x 1,2 where F = 0. Next, we note that Equation (19) can be formally integrated once to yield…”
Section: Integrabilitymentioning
confidence: 99%
See 1 more Smart Citation
“…However, the full equation is not integrable, with a breakdown on the lines X = x 1,2 where F = 0. Next, we note that Equation (19) can be formally integrated once to yield…”
Section: Integrabilitymentioning
confidence: 99%
“…The reduced Ostrovsky Equation (4) (also known variously as the Ostrovsky-Hunter equation, or the Vakhnenko equation) has been previously studied numerically and theoretically, notably by Hunter [15] Vakhnenko [23], Parkes [21], Vakhnenko and Parkes [24], Boyd [1], [2], Stepanyants [22], Liu et al [19], and Kraenkel et al [16]. We also note that it is readily shown that Equation (4) does not have any smooth solitary wave solutions (see Liu et al [19] and the appendix of Grimshaw and Helfrich [10], where the argument produced there still applies when λ = 0), but does support a family of smooth periodic traveling wave solutions (Ostrovsky,[20]). Thus, clearly not all solutions will break, although Boyd [2] showed numerically that a large class of solutions will break, even when the amplitude is very small provided that the length scale is correspondingly also very small.…”
Section: Introductionmentioning
confidence: 99%
“…Let > 0. Then, • For 1 < < 3, the constrained minimization problems (9) and (10) have solutions and . In addition, ∈ 2 ∩̇− 2 (ℝ), ∈ 4 (ℝ) ∶ ′ = and they satisfy, for some = ∈ (−∞, 2), 2 + −2 + + | | = 0,…”
Section: Existence Of the Normalized Wavesmentioning
confidence: 99%
“…Assuming that the time derivative is treated locally (i.e., as the usual partial derivative ∂/∂t), there are two main aspects of the general advection equation for which nonlocal extensions have been examined: (i) nonlocal f , nonlocal f x (u) in (2.0.2a), or nonlocal f (u) in (2.0.2b), and (ii) nonlocal regularizations, i.e., "small" nonlocal terms replacing the zero on the RHS of the equality in both equations in (2.0.2). The first of these approaches was considered, e.g., in [6,9,14,38,56,66], while the second category of nonlocalizations includes some of the above cited works, as well as, e.g., [1,2,4,7,11,15,16,19,28,27,36,37,56,49,63,64]. A focus of many of these works is on nonlocal generalizations of Burgers equation.…”
Section: Previous Approaches To Non-local Advectionmentioning
confidence: 99%