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Concept# Integrable system

Summary

In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold
of much smaller dimensionality than that of its phase space.
Three features are often referred to as characterizing integrable systems:

- the existence of a maximal set of conserved quantities (the usual defining property of complete integrability)
- the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability)
- the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability)

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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 not equal 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.

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar, equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler-Poincar, equations that have not yet been considered in the literature as well as integrable equations like Camassa-Holm, Degasperis-Procesi, mu CH and mu DP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle.

A class of Neumann type systems are derived separating the spatial and temporal variables for the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation and the modified Korteweg-de Vries (mKdV) hierarchy. The Lax-Moser matrix of Neumann type systems is worked out, which generates a sequence of integrals of motion and a hyperelliptic curve of KdV type. We deduce the constrained Hamiltonians to put Neumann type systems into canonical Hamiltonian equations and further complete the Liouville integrability for the Neumann type systems. We also specify the relationship between Neumann type systems and infinite dimensional integrable systems (IDISs), where the involutivity solutions of Neumann type systems yield the finite parametric solutions of IDISs. From the Abel-Jacobi variables, the evolution behavior of Neumann type flows are shown on the Jacobian of a Riemann surface. Finally, the Neumann type flows are applied to produce some explicit solutions expressed by Riemann theta functions for the 2+1 dimensional CDGKS equation and the mKdV hierarchy.

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