17 research outputs found
Smooth and Peaked Solitons of the CH equation
The relations between smooth and peaked soliton solutions are reviewed for
the Camassa-Holm (CH) shallow water wave equation in one spatial dimension. The
canonical Hamiltonian formulation of the CH equation in action-angle variables
is expressed for solitons by using the scattering data for its associated
isospectral eigenvalue problem, rephrased as a Riemann-Hilbert problem. The
momentum map from the action-angle scattering variables
to the flow momentum () provides the Eulerian representation of
the -soliton solution of CH in terms of the scattering data and squared
eigenfunctions of its isospectral eigenvalue problem. The dispersionless limit
of the CH equation and its resulting peakon solutions are examined by using an
asymptotic expansion in the dispersion parameter. The peakon solutions of the
dispersionless CH equation in one dimension are shown to generalize in higher
dimensions to peakon wave-front solutions of the EPDiff equation whose
associated momentum is supported on smoothly embedded subspaces. The Eulerian
representations of the singular solutions of both CH and EPDiff are given by
the (cotangent-lift) momentum maps arising from the left action of the
diffeomorphisms on smoothly embedded subspaces.Comment: First version -- comments welcome! Submitted to JPhys
Similarity reductions of peakon equations: integrable cubic equations
We consider the scaling similarity solutions of two integrable cubically
nonlinear partial differential equations (PDEs) that admit peaked soliton
(peakon) solutions, namely the modified Camassa-Holm (mCH) equation and
Novikov's equation. By making use of suitable reciprocal transformations, which
map the mCH equation and Novikov's equation to a negative mKdV flow and a
negative Sawada-Kotera flow, respectively, we show that each of these scaling
similarity reductions is related via a hodograph transformation to an equation
of Painlev\'e type: for the mCH equation, its reduction is of second order and
second degree, while for Novikov's equation the reduction is a particular case
of Painlev\'e V. Furthermore, we show that each of these two different
Painlev\'e-type equations is related to the particular cases of Painlev\'e III
that arise from analogous similarity reductions of the Camassa-Holm and the
Degasperis-Procesi equation, respectively. For each of the cubically nonlinear
PDEs considered, we also give explicit parametric forms of their periodic
travelling wave solutions in terms of elliptic functions. We present some
parametric plots of the latter, and, by using explicit algebraic solutions of
Painlev\'e III, we do the same for some of the simplest examples of scaling
similarity solutions, together with descriptions of their leading order
asymptotic behaviour