G.R.W. Quispel

Integrable mappings and soliton equations II

Abstract Some simple solutions of discrete soliton equations are shown to satisfy 2D mappings. We show that these belong to a recently introduced 18-parameter family of integrable reversible mappings of the plane, thus lending weight to a previous conjecture. We also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics. Finally we discuss the notion of (generalized) reversibility.

Geometric integration using discrete gradients

This paper discusses the discrete analogue of the gradient of a function and shows how discrete gradients can be used in the numerical integration of ordinary differential equations (ODEs). Given an ODE and one or more first integrals (i.e. constants of the motion) and/or Lyapunov functions, it is shown that the ODE can be rewritten as a ‘linear–gradient system’. Discrete gradients are used to construct discrete approximations to the ODE which preserve the first integrals and Lyapunov functions exactly. The method applies to all Hamiltonian, Poisson and gradient systems, and also to many dissipative systems (those with a known first integral or Lyapunov function).

Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models

Eigenspectra of the critical quantum Ashkin-Teller and Potts chains with free boundaries can be obtained from that of the XXZ chain with free boundaries and a complex surface field. By deriving and solving numerically the Bethe ansatz equations for such boundaries the authors obtain eigenenergies of XXZ chains of up to 512 sites. The conformal anomaly and surface exponents of the quantum XXZ, Ashkin-Teller, and Potts chains are calculated by exploiting their relations with the mass gap amplitudes as predicted by conformal invariance.

Integrable mappings and soliton equations

Abstract We report an 18-parameter family of integrable reversible mappings of the plane. These mappings are shown to occur in soliton theory and in statistical mechanics. We conjecture that all autonomous reductions of differential-difference soliton equations are integrable mappings.

Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems

Abstract Dynamical systems with independent (continuous or discrete) time variable t and phase space variable x are called reversible if they are invariant under the combination { t →− t , x → G x } where G is some transformation of phase space which is an involution ( G ∘ G = Identity ). Reversible systems generalise classical mechanical systems possessing time-reversal symmetry and are found in ordinary differential equations, partial differential equations and diffeomorphisms (mappings) modelling many physical problems. This report is an introduction to some of the properties of reversible systems, with particular emphasis on reversible mappings of the plane which illustrate many of their basic features. Reversible dynamical systems are shown to be similar to Hamiltonian systems because they can possess KAM tori, yet they are different because they can also have attractors and repellers. We create and study examples of these hybrid dynamical systems and discuss the question of how to recognise whether a given dynamical system is reversible.

A new class of energy-preserving numerical integration methods

The first ever energy-preserving B-series numerical integration method for (ordinary) differential equations is presented and applied to several Hamiltonian systems. Related novel Lie algebraic results are also discussed.

Direct linearization of nonlinear difference-difference equations

Abstract Starting from the linear integral equation for the solutions of the Korteweg-de Vries (KdV) equation, we obtain the direct linearization of a general nonlinear difference-difference equation. In a continuum limit this equation reduces to a general integrable differential-difference equation which contains e.g. the Toda equation and the discrete KdV and MKdV as special cases.

Linear integral equations and nonlinear difference-difference equations

In this paper we present a systematic method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred. It is argued that these difference-difference equations can be regarded as arising from Bianchi identities expressing the commutativity of Backlund transformations. Applying an appropriate continuum limit we first obtain integrable nonlinear differential-difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations. As special cases we treat the difference-difference versions and the differential-difference versions of the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schrodinger equation, the isotropic classical Heisenberg spin chain, and the complex and real sine-Gordon equation.

Preserving energy resp. dissipation in numerical PDEs using the Average Vector Field method

We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine-Gordon, Korteweg-de Vries, nonlinear Schrodinger, (linear) time-dependent Schrodinger, and Maxwell equations. In the dissipative case the examples are: the Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, and heat equations.

Integrable mappings derived from soliton equations

We derive a hierarchy of ibtegrable mappings (integrable ordinary difference equations) corresponding to solutions of the initial-value problem of an integrable partial difference equation with periodic initial data. For each n ϵ N this hierarchy contains at least one integrable mapping Rn→Rn. The integrals of these mappings are constructed using the Lax pair of the underlying partial difference equation. Our approach is illustrated for the integrable partial difference analogues of the sine-Gordon and the (modified) Korteweg-de Vries equations.