The two-component Camassa-Holm system models the propagation of unidirectional shallow water waves over a flat bed and, in some instances, admits an interpretation as a geodesic equation for a right-invariant metric on the diffeomorphism group.
In this talk, a short-wave model for this system is studied with breakdown (or global existence in time) of classical solutions being discussed.
On a short-wave model for the two-component Camassa-Holm system
In his paper Northcott's Theorem on Heights II. The Quadratic Case, W.M. Schmidt gave asymptotic estimates for the number of points in projective space that have height no greater than a given bound and that generate a quadratic extension of the rational number field. For the special case where the points lie in projective 2-space, he needed to prove a result on sums of quotients of certain L-series (coming from quadratic extensions of the rational number field). Motivated by the function field case of Schmidt's result, we will look at L-series over function fields and discuss some recent results.
On a short-wave model for the two-component Camassa-Holm system