Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

A new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method. The method proposed allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian. The general considerations are applied to the XXZ and XYZ models, the nonlinear Schrodinger equation and Toda chain.

https://math.berkeley.edu/~reshetik/Sklyanin-boundary.pdf

Boundary conditions for integrable quantum systems

This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented.

/pdf/discrete-mechanics-and-variational-integrators-4qm2t5dk6u.pdf

Discrete mechanics and variational integrators

https://digital.csic.es/bitstream/10261/15682/1/blanes.pdf

The Magnus expansion and some of its applications

Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented. These new similarity reductions, including some new reductions to the first, second, and fourth Painleve equations, cannot be obtained using the standard Lie group method for finding group‐invariant solutions of partial differential equations; they are determined using a new and direct method that involves no group theoretical techniques.

New similarity reductions of the Boussinesq equation

In recent work it was shown how recursive factorisation of certain QRT maps leads to Somos-4 and Somos-5 recurrences with periodic coefficients, and to a fifth-order recurrence with the Laurent property. Here we recursively factorise the 12-parameter symmetric QRT map, given by a second-order recurrence, to obtain a system of three coupled recurrences which possesses the Laurent property. As degenerate special cases, we first derive systems of two coupled recurrences corresponding to the 5-parameter multiplicative and additive symmetric QRT maps. In all cases, the Laurent property is established using a generalisation of a result due to Hickerson, and exact formulae for degree growth are found from ultradiscrete (tropical) analogues of the recurrences. For the general 18-parameter QRT map it is shown that the components of the iterates can be written as a ratio of quantities that satisfy the same Somos-7 recurrence.

QRT maps and related Laurent systems

Manin transformations are maps of the plane that preserve a pencil of cubic curves. They are the composition of two involutions. Each involution is constructed in terms of an involution point that is required to be one of the base points of the pencil. We generalise this construction to explicit birational maps of the plane that preserve quadratic resp. certain quartic pencils, and show that they are measure-preserving and hence integrable. In the quartic construction the two involution points are required to be base points of the pencil of multiplicity 2. On the other hand, for the quadratic pencils the involution points can be any two distinct points in the plane (except for base points). We employ Pascal's theorem to show that the maps that preserve a quadratic pencil admit infinitely many symmetries. The full 18-parameter QRT map is obtained as a special instance of the quartic case in a limit where the two involution points go to infinity. We show by construction that each generalised Manin transformation can be brought to QRT form by a fractional affine transformation. We also specify classes of generalised Manin transformations which admit a root.

Generalised Manin transformations and QRT maps

It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry transformation and apply this idea to autonomous and non-autonomous lattice equations. As results of this approach, we obtain new reductions of the discrete potential Korteweg–de Vries (KdV) equation, discrete modified KdV equation and the discrete Schwarzian KdV equation. We will also describe a direct method for obtaining Lax representations for the reduced equations.

/pdf/twisted-reductions-of-integrable-lattice-equations-and-their-5azol63o0t.pdf

Twisted reductions of integrable lattice equations, and their Lax representations

We prove the Liouville and superintegrability of a generalized Lotka–Volterra system and its Kahan discretization.

/pdf/liouville-integrability-and-superintegrability-of-a-6t6ava5jiq.pdf

Liouville integrability and superintegrability of a generalized Lotka–Volterra system and its Kahan discretization

https://dspace.library.uu.nl/bitstream/handle/1874/877/c2.pdf

G. R. W. Quispel

Papers

QRT maps and related Laurent systems

Generalised Manin transformations and QRT maps

Twisted reductions of integrable lattice equations, and their Lax representations

Liouville integrability and superintegrability of a generalized Lotka–Volterra system and its Kahan discretization

Geometric numerical integration applied to the elastic pendulum at higher-order resonance