Abstract
The celebrated Toda lattice was originally obtained as a simple model for describing a chain of particles with nearest neighbour exponential interaction and has been generalized in multiple ways. A class of nonlinear integrable partial differential equations (PDEs) admit some special weak solutions called “peakons”, which are characterised by systems of ordinary differential equations (ODEs), namely peakon lattices. It is shown that Toda and peakon type lattices can be regarded as isospectral deformations in opposite directions related to certain orthogonal functions. We will take the ordinary Toda lattice and the Camassa-Holm peakons, the Kac-van Moerbeke lattice and a two-component modified Camassa-Holm interlacing peakons, the B-Toda lattice and the Novikov peakons, the C-Toda lattice and the Degasperis-Procesi peakons, the Frobenius-Stickelberger-Thiele lattice and the modified Camassa-Holm peakons as examples to illustrate this point of view.
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