H.W. Capel

The Discrete Korteweg—de Vries Equation

We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlevé equations as well as connections to Volterra systems.

Integrable mappings and nonlinear integrable lattice equations

Abstract Periodic initial value problems of time and space discretizations of integrable partial differential equations give rise to multi-dimensional integrable mappings. Using the associated linear spectral problems (Lax pairs), a systematic derivation is given of the corresponding sets of polynomial invariants. The level sets are algebraic varieties on which the trajectories of the corresponding dynamical systems lie.

COMPLETE INTEGRABILITY OF LAGRANGIAN MAPPINGS AND LATTICES OF KDV TYPE

Abstract We present the Lagrangian and (time-discrete) Hamiltonian structures of lattice discretizations of the KdV equation, as well as of the associated finite-dimensional mappings that we derived earlier. Complete integrability in the sense of Liouville of these mappings is established by showing involutivity of a complete set of integrals of the discrete-time dynamics. Similar results hold for lattices and mappings related to the MKdV and Toda 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.

Bäcklund transformations and three-dimensional lattice equations

A (nonlocal) linear integral equation is studied, which allows for Backlund transformations in the measure. The compatibility of three of these transformations leads to an integrable nonlinear three-dimensional lattice equation. In appropriate continuum limits the two-dimensional Toda-lattice equation and the Kadomtsev-Petviashvili equation are derived as special examples.

Linearization of the Boussinesq equation and the modified Boussinesq equation

Abstract A description in terms of one and the same inhomogeneous linear integral equation is proposed for the solutions of the Boussinesq equation and the modified Boussinesq equation. New similarity solutions of these equations are obtained, as well as two-parameter families of solutions of Painleve II and Painleve IV.

Solving ODEs numerically while preserving a first integral

Abstract We give general algorithms for the numerical integration of ordinary differential equations (ODEs) that possess a first integral I(x). Our discrete algorithms preserve the integral I exactly. Our method works both for dissipative and for all Hamiltonian ODEs. For non-Hamiltonian systems a sufficient (but not necessary) condition for our method to work is that ƒ∇Iƒ ≠ 0 on the isosurface we are integrating on.

On some linear integral equations generating solutions of nonlinear partial differential equations

Two types of linear inhomogeneous integral equations, which yield solutions of a broad class of nonlinear evolution equations, are investigated. One type is characterized by a two-fold integration with an arbitrary measure and contour over a complex variable k, and thier complex conjugates, whereas the other one has a two-fold integration over one and the same contour. The inhomogeneous term, which may contain an arbitrary function of k, makes it possible to define a matrix structure on the solutions of the integral equations. The elements of these matrices are shown to obey a system of partial differential equations, the special form of which depends on the choice of the dispersion relation occurring in the integral equations. For special elements of the matrices closed partial differential equations can be derived, such as e.g. the nonlinear Schrodinger equation and the (real and complex) modified Korteweg-de Vries and sine-Gordon equations. The relations between the matrix elements are shown to lead to Miura transformations between the various partial differential equations.

Multibreathers and homoclinic orbits in 1-dimensional nonlinear lattices

Spatially localized, time-periodic excitations, known as discrete breathers, have been found to occur in a wide variety of 1-dimensional (1-D) lattices of nonlinear oscillators with nearest-neighbour coupling. Eliminating the time-dependence from the differential-difference equations of motion, and taking into account only the N largest Fourier modes, we view these solutions as orbits of a (non-integrable) 2N-D map. For breathers to occur, the trivial rest state of the lattice must be a hyperbolic fixed point of the map, with an N-D stable and an N-D unstable manifold. The breathers and multibreathers (with one and more spatial oscillations respectively) are then directly related to the intersections of these manifolds, and hence to homoclinic orbits of the 2N-D map. This is explicitly shown here on a discretized nonlinear Schrodinger equation with only one Fourier mode (N=1), represented by a 2-D map. We then construct the 2N-D map for an array of nonlinear oscillators, with nearest neighbour coupling and a quartic on-site potential, and demonstrate how a one-Fourier-mode representation (via a 2-D map) can be used to provide remarkably accurate initial conditions for the breather and multibreather solutions of the system.

k-integrals and k-Lie symmetries in discrete dynamical systems

We generalize the concept of symplectic maps to that of k- symplectic maps: maps whose kth iterates are symplectic. Similarly, k-symmetries and k-integrals are symmetries (resp. integrals) of the kth iterate of the map. It is shown that k-symmetries and k-integrals are related by the k-symplectic structure, as in the k = 1 continuous case (Noethers theorem). Examples are given of k-integrals and their related k-symmetries for k = 1,…,4.