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

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