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

Euler-Poincar\'e equations for GG-Strands

The $G$-strand equations for a map $\mathbb{R}\times \mathbb{R}$ into a Lie group $G$ are associated to a $G$-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The $G$-strand itself is the map $g(t,s): \mathbb{R}\times \mathbb{R}\to G$, where $t$ and $s$ are the independent variables of the $G$-strand equations. The Euler-Poincar\'e reduction of the variational principle leads to a formulation where the dependent variables of the $G$-strand equations take values in the corresponding Lie algebra $\mathfrak{g}$ and its co-algebra, $\mathfrak{g}^*$ with respect to the pairing provided by the variational derivatives of the Lagrangian. We review examples of different $G$-strand constructions, including matrix Lie groups and diffeomorphism group. In some cases the $G$-strand equations are completely integrable 1+1 Hamiltonian systems that admit soliton solutions.Comment: To appear in Conference Proceedings for Physics and Mathematics of Nonlinear Phenomena, 22 - 29 June 2013, Gallipoli (Italy) http://pmnp2013.dmf.unisalento.it/talks.shtml, 9 pages, no figure