The Wave Breaking for Whitham-Type Equations Revisited

Saut, Jean‐Claude; Wang, Yuexun

doi:10.1137/20m1345207

Cited by 12 publications

(9 citation statements)

References 19 publications

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“…can form shocks for large solutions in the range α ∈ (−1, −1/3) [9,10,30] and this was recently extended to the whole interval (−1, 0) [26]. We will show a similar shock formation result holds true for the equation (1.1) in the whole range α ∈ (−1, 0).…”

Section: Proposition 11 ([29]supporting

confidence: 67%

“…The possibility of appearance of shocks was proven in [9,10] when −1 < α < −1/3 and for the Whitham equation. A simpler proof, applying to the case −1 < α < −2/5 and to the Whitham equation was given in [30]. The shock formation in the full range −1 < α < 0 was recently established in [26].…”

Section: Remark 23mentioning

confidence: 99%

“…The following technical lemmas are a variant of those in [9,25,30] which dealt with the quadratic nonlinearity. Differently, here we need to work near the shock to deal with the cubic nonlinearity.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…x u L ∞ in order to estimate K 1 (t, x) than that of the fKdV equation due to the stronger nonlinear effect present in the equation (1.1). The proof on K 0 (t, x) is the same as that of [30], we include it here for sake of completeness. To estimate the terms K 0 (t, x) and K 1 (t, x), we perform the following decompositions…”

Section: Estimates On Nonlocal Termsmentioning

confidence: 99%

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On the modified fractional Korteweg–de Vries and related equations

2022

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We consider in this paper modified fractional Korteweg–de Vries and related equations (modified Burgers–Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low frequencies and tends to zero at infinity and the strongly dispersive case where the phase velocity vanishes at the origin and goes to infinity at infinity. In the former case, the nonlinear hyperbolic effects dominate for large data, leading to the possibility of shock formation though the dispersive effects manifest for small initial data where scattering is possible. In the latter case, finite time blow-up is possible in the focusing case but not the shock formation. In the defocusing case global existence and scattering is expected in the energy subcritical case, while finite time blow-up is expected in the energy supercritical case. We establish rigorously the existence of shocks with blow-up time and location being explicitly computed in the weakly dispersive case, while most of the results on the strongly dispersive case are derived via numerical simulations, for large solutions. Moreover, the shock formation result can be extended to the weakly dispersive equation with some generalized nonlinearity. We will also comment briefly on the BBM versions of those equations.

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Section: Proposition 11 ([29]supporting

confidence: 67%

Section: Remark 23mentioning

confidence: 99%

Section: Proof Of Theorem 21mentioning

confidence: 99%

Section: Estimates On Nonlocal Termsmentioning

confidence: 99%

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On the modified fractional Korteweg–de Vries and related equations

2022

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“…where φ represents the surface profile and KW (ξ) = ˆR K W (x)e −ixξ dx := tanh(ξ) ξ , is a fully dispersive variant of the classical Korteweg-de Vries (KdV) equation, originally proposed in [35]. It features some properties that the KdV equation lacks, such as wave breaking [23,32], highest waves [13,15,34], and better high-frequency modelling [17]. While Whitham added the dispersion in an ad hoc manner, the model has since been both justified experimentally and derived from the full water-wave problem in several ways.…”

Section: Introductionmentioning

confidence: 99%

On the precise cusped behaviour of extreme solutions to Whitham-type equations

Ehrnström¹,

Mæhlen²,

Varholm³

2023

Preprint

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We prove exact leading-order asymptotic behaviour at the origin for nontrivial solutions of two families of nonlocal equations. The equations investigated include those satisfied by the cusped highest steady waves for both the uni-and bidirectional Whitham equations. The problem is therefore analogous to that of capturing the 120°interior angle at the crests of classical Stokes' waves of greatest height. In particular, our results partially settle conjectures for such extreme waves posed in a series of recent papers [14,15,34]. Our methods may be generalised to solutions of other nonlocal equations, and can moreover be used to determine asymptotic behaviour of their derivatives to any order.

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Analysis on continuity of the solution map for the Whitham equation in Besov spaces

Liu,

2023

Mathematische Nachrichten

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This paper is devoted to studying the continuity of the solution map for the Cauchy problem of the Whitham equation. First, the continuity dependence of solution is established in in the sense of Hadamard. Next, by constructing approximate solutions, we show that the data‐to‐solution map is not uniformly continuous in Besov spaces () on the periodic case and on the line. The crucial technical tool used to prove this result is Lemma 5.1, which generalizes the result of Lemma 3 in [23].

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The Wave Breaking for Whitham-Type Equations Revisited

Cited by 12 publications

References 19 publications

On the modified fractional Korteweg–de Vries and related equations

On the modified fractional Korteweg–de Vries and related equations

On the precise cusped behaviour of extreme solutions to Whitham-type equations

Analysis on continuity of the solution map for the Whitham equation in Besov spaces

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