Weak solution of the Novikov equation and optimal control

Shen, Chunyu

doi:10.1016/j.ejcon.2019.06.004

Cited by 2 publications

(1 citation statement)

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“…The literature surrounding the Novikov equation is not yet quite as overwhelming as for the CH and DP equations, but there is no shortage of articles about PDE-analytic questions, in addition to the one just mentioned [253,290,305,176,147,314,306,135,150,195,315,196,307,53,304,143,34,149,328,302,63,228,278,207]. Stability of peakons has been considered by several researchers [215,299,257,258,54,55], and likewise solitons [237,206,260,259,301,229,326] and integrability aspects [281,30,186,29,273].…”

Section: The Novikov Equationmentioning

confidence: 99%

A view of the peakon world through the lens of approximation theory

Lundmark,

Szmigielski

2022

Preprint

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Peakons (peaked solitons) are a particular type of solution admitted by certain nonlinear PDEs, most famously the Camassa-Holm shallow water wave equation. These solutions take the form of a train of peak-shaped waves, interacting in a particle-like fashion. In this article we give an overview of the mathematics of peakons, with special emphasis on the connections to classical problems in analysis, such as Padé approximation, mixed Hermite-Padé approximation, multi-point Padé approximation, continued fractions of Stieltjes type and (bi)orthogonal polynomials. The exposition follows the chronological development of our understanding, exploring the peakon solutions of the Camassa-Holm, Degasperis-Procesi, Novikov, Geng-Xue and modified Camassa-Holm (FORQ) equations. All of these paradigm examples are integrable systems arising as the compatibility condition of a Lax pair, and a recurring theme in the context of peakons is the need to properly interpret these Lax pairs in the sense of Schwartz's theory of distributions. We trace out the path leading from distributional Lax pairs to explicit formulas for peakon solutions via a multitude of approximation-theoretic problems, and illustrate the dynamics of peakons with graphics. ContentsThe modified Camassa-Holm equation and distributional Lax integrability 31

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