Extended Camassa-Holm Hierarchy and Conserved Quantities

An extension of the Camassa-Holm hierarchy is constructed in this letter. The conserved quantities of the hierarchy are studied and a recurrent formula for the integrals of motion is derived.Comment: 13 page

Hamiltonian formulation and integrability of a complex symmetric nonlinear system

The integrability of a complex generalisation of the 'elegant' system, proposed by D. Fairlie and its relation to the Nahm equation and the Manakov top is discussed.Comment: 8 pages, Physics Letters A (accepted

Empirical balanced truncation of nonlinear systems

Novel constructions of empirical controllability and observability gramians for nonlinear systems for subsequent use in a balanced truncation style of model reduction are proposed. The new gramians are based on a generalisation of the fundamental solution for a Linear Time-Varying system. Relationships between the given gramians for nonlinear systems and the standard gramians for both Linear Time-Invariant and Linear Time-Varying systems are established as well as relationships to prior constructions proposed for empirical gramians. Application of the new gramians is illustrated through a sample test-system.Comment: LaTeX, 11 pages, 2 figure

Integrable models for shallow water with energy dependent spectral problems

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependance on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup-Bousinesq equation. We formulate the inverse problem as a Riemann-Hilbert problem with a Z2 reduction group. The soliton solutions are explicitly obtained.Comment: 17 pages, 3 figure

Poisson structure and Action-Angle variables for the Camassa-Holm equation

The Poisson brackets for the scattering data of the Camassa-Holm equation are computed. Consequently, the action-angle variables are expressed in terms of the scattering data.Comment: 20 pages, LaTeX. The original publication is available at www.springerlink.co

Some Irrational Generalised Moonshine from Orbifolds

We verify the Generalised Moonshine conjectures for some irrational modular functions for the Monster centralisers related to the Harada-Norton, Held, $M_{12}$ and $L_3(3)$ simple groups based on certain orbifolding constraints. We find explicitly the fixing groups of the hauptmoduls arising in each case