R.G. Halburd∗ Department of Mathematical Sciences, Loughborough University Loughborough, Leicestershire, LE11 3TU, UK (Dated: January 21, 2005) Abstract The heights of iterates of the discrete Painlevé equations over number fields appear to grow no faster than polynomials while the heights of generic solutions of non-integrable discrete equations grow exponentially. This gives rise to a simple ...

This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between th...

This paper discusses the algorithms and implementations of three Mathematica packages for the study of integrability and the computation of closed-form solutions of nonlinear polynomial PDEs. The first package, PainleveTest.m, symbolically performs the Painlevé integrability test. The second package, PDESpecialSolutions.m, computes exact solutions expressible in hyperbolic or elliptic functions...

In this paper we discuss local and formal Darboux first integrals of analytic differential systems, using the theory of PoincaréDulac normal forms. We study the effect of local Darboux integrability on analytic normalization. Moreover we determine local restrictions on classical Darboux integrability of polynomial systems.

The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define Weierstrass integrability and we determine which Weierstrass integrable systems are Liouvillian integrable. Inside this new class of inte...

It is shown how functional-analytic gradient-holonomic structures can be used for an isospectral integrability analysis of nonlinear dynamical systems on discrete manifolds. The approach developed is applied to obtain detailed proofs of the integrability of the discrete nonlinear Schrödinger, Ragnisco–Tu and Riemann–Burgers dynamical systems.

For one dimensional SU(n) Hubbard model, a pair of Lax operators are derived, which give a set of fundamental equations for the quantum inverse scattering method under both periodic and open boundary conditions. This provides another proof of the integrability of the model under periodic boundary condition. PACS Numbers: 46.10.+z, 05.40.+j, 05.60.+w e-mail: [email protected] Integrable strong...

Consider a K-principal fibre bundle P π −→ M and fix a connection A on P . Suppose an ad(P )-valued 1-form α defines a bundle isomorphism TM ∼= −→ ad(P ). We will associate to any such triple (π,A, α) a K-invariant almost complex structure Jα on P . In the first section we give explicitely the integrability condition of Jα. This condition is equivalent to a system on non-linear differential equ...