Integrable evolution equations on spaces of tensor densities and their peakon solutions

TL;DR

This paper introduces two new integrable PDEs on tensor density spaces, explores their mathematical structures, and constructs explicit peakon solutions, expanding understanding of integrable systems with geometric and physical relevance.

Contribution

It presents two novel integrable equations related to tensor densities, analyzes their structures, and constructs explicit peakon solutions for various parameter values.

Findings

Two new integrable PDEs on tensor densities are introduced.

Explicit peakon and shock-peakon solutions are constructed.

The equations are shown to have Lax pairs and bihamiltonian structures.

Abstract

We study a family of equations defined on the space of tensor densities of weight $\lambda$ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct "peakon" and "multi-peakon" solutions for all $\lambda \neq 0,1$, and "shock-peakons" for $\lambda = 3$. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold's approach to Euler equations on Lie groups.