Dispersionless Camassa-Holm Equation Research Articles

A coupling problem for entire functions and its application to the long-time asymptotics of integrable wave equations

We propose a novel technique for analysing the long-time asymptotics of integrable wave equations in the case when the underlying isospectral problem has a purely discrete spectrum. To this end, we introduce a natural coupling problem for entire functions, which serves as a replacement for the usual Riemann–Hilbert problem, which does not apply in these cases. As a prototypical example, we investigate the long-time asymptotics of the dispersionless Camassa–Holm equation.

An asymptotic property of the Camassa–Holm equation

Asymptotic densities are used to study an asymptotic property of the dispersionless Camassa–Holm equation. An asymptotic density of a global solution is a weak limit of its scaled momentum density along a sequence of time increasing to infinity. For a global solution with non-negative compactly supported initial momentum density, we show that if the asymptotic density is unique, then it is a positive combination of Dirac measures supported in a bounded interval in the non-negative axis with zero as the only possible accumulation point. In other words, if the scaled momentum density does not oscillate as time goes to infinity, the solution behaves as a combination of peakons moving to the right at different speeds. In contrast to many investigations on the topic, our approach is not spectral theoretic and hence is independent of the structure of the isospectral problem associated with the equation.

On the isospectral problem of the dispersionless Camassa–Holm equation

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.

Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions

The EPDiff equation (or the dispersionless Camassa–Holm equation in one dimension) is a well-known example of geodesic motion on the Diff group of smooth invertible maps (diffeomorphisms). Its recent two-component extension governs geodesic motion on the semidirect product DiffⓈ , where denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on DiffⓈ , where denotes the space of scalar functions that take values on a certain Lie algebra (e.g. = ⊗ (3)). Measure-valued delta-like solutions are shown to be momentum maps possessing a dual pair structure, thereby extending previous results for the EPDiff equation. The collective Hamiltonians are shown to fit into the Kaluza–Klein theory of particles in a Yang–Mills field and these formulations are shown to apply also at the continuum partial differential equation level. In the continuum description, the Kaluza–Klein approach produces the Kelvin circulation theorem.

A Class of Equations with Peakon and Pulson Solutions (with an Appendix by Harry Braden and John Byatt-Smith)

We consider a family of integro-differential equations depending upon a parameter b as well as a symmetric integral kernel g(x). When b = 2 and g is the peakon kernel (i.e. g(x) = exp(−|x|) up to rescaling) the dispersionless Camassa-Holm equation results, while the Degasperis-Procesi equation is obtained from the peakon kernel with b = 3. Although these two cases are integrable, generically the corresponding integro-PDE is non-integrable. However, for b = 2 the family restricts to the pulson family of Fringer & Holm, which is Hamiltonian and numerically displays elastic scattering of pulses. On the other hand, for arbitrary b it is still possible to construct a nonlocal Hamiltonian structure provided that g is the peakon kernel or one of its degenerations: we present a proof of this fact using an associated functional equation for the skew-symmetric antiderivative of g. The nonlocal bracket reduces to a non-canonical Poisson bracket for the peakon dynamical system, for any value of b ≠ 1.