Symmetric calorons (2004)
Journal Article
Ward, R. (2004). Symmetric calorons. Physics Letters B, 582(3-4), 203-210. https://doi.org/10.1016/j.physletb.2003.12.051

Calorons (periodic instantons) interpolate between monopoles and instantons, and their holonomy gives approximate Skyrmion configurations. We show that, for each caloron charge N4, there exists a one-parameter family of calorons which are symmetric u... Read More about Symmetric calorons.

Topological Q-solitons. (2003)
Journal Article
Ward, R. (2003). Topological Q-solitons. Journal of Mathematical Physics, 44, 3555-3561

Stabilizing textures with magnetic fields (2002)
Journal Article
Ward, R. (2002). Stabilizing textures with magnetic fields. Physical Review D, Particles and fields, 66(4), Article 041701(R). https://doi.org/10.1103/physrevd.66.041701

The best-known way of stabilizing textures is by Skyrme-like terms, but another possibility is to use gauge fields. The semilocal vortex may be viewed as an example of this, in two spatial dimensions. In three dimensions, however, the idea (in its si... Read More about Stabilizing textures with magnetic fields.

Stability of sigma-model strings and textures. (2002)
Journal Article
Ward, R. (2002). Stability of sigma-model strings and textures. Classical and Quantum Gravity, 19(4), L17-L22. https://doi.org/10.1088/0264-9381/19/4/101

In flat spacetime, sigma-model strings and textures are both unstable: they tend to collapse (and subsequently decay). With sufficient cosmological expansion, however, they are stable in a generalized sense: a small perturbation will cause them to ch... Read More about Stability of sigma-model strings and textures..

Hopf solitons from instanton holonomy. (2001)
Journal Article
Ward, R. (2001). Hopf solitons from instanton holonomy. Nonlinearity, 14(6), 1543-1554. https://doi.org/10.1088/0951-7715/14/6/307

The holonomy of an SU(2) N-instanton in the x4-direction gives a map from Bbb R3 into SU(2), which provides a good model of an N-Skyrmion. Combining this map with the standard Hopf map from SU(2)congS3 to S2 gives a configuration for a Hopf soliton o... Read More about Hopf solitons from instanton holonomy..

Skyrmions on the 2-sphere. (2001)
Journal Article
de Innocentis, M., & Ward, R. (2001). Skyrmions on the 2-sphere. Nonlinearity, 14(3), 663-671. https://doi.org/10.1088/0951-7715/14/3/312

We study static solutions of the Skyrme model on the 2-sphere of radius L, for various choices of potential. The high-density Skyrmion phase corresponds to the ratio β = L/(size of Skyrmion) being small, whereas the low-density phase corresponds to β... Read More about Skyrmions on the 2-sphere..

Integrable Yang-Mills-Higgs equations in 3-dimensional de Sitter space-time. (2001)
Journal Article
Kotecha, V., & Ward, R. (2001). Integrable Yang-Mills-Higgs equations in 3-dimensional de Sitter space-time. Journal of Mathematical Physics, 42(3), 1018-1025. https://doi.org/10.1063/1.1345499

This paper describes an integrable Yang–Mills–Higgs system on (2+1)-dimensional de Sitter space–time. It is the curved-space–time analog of the Bogomolnyi equations for monopoles on R3.R3. A number of solutions, of various types, are constructed. Read More about Integrable Yang-Mills-Higgs equations in 3-dimensional de Sitter space-time..

The interaction of two hopf solitons. (2000)
Journal Article
Ward, R. (2000). The interaction of two hopf solitons. Physics Letters B, 473(3-4), 291-296. https://doi.org/10.1016/s0370-2693%2899%2901503-8

This Letter deals with topological solitons in an O(3) sigma model in three space dimensions (with a Skyrme term to stabilize their size). The solitons are classified topologically by their Hopf number N. The N=2 sector is studied; in particular, for... Read More about The interaction of two hopf solitons..

Two integrable systems related to hyperbolic monopoles. (1999)
Journal Article
Ward, R. (1999). Two integrable systems related to hyperbolic monopoles. Asian Journal of Mathematics, 3(1), 325-332. https://doi.org/10.4310/ajm.1999.v3.n1.a12

Monopoles on hyperbolic 3-space were introduced by Atiyah in 1984. This article describes two integrable systems which are closely related to hyperbolic monopoles: a one-dimensional lattice equation (the Braam-Austin or discrete Nahm equation), and a... Read More about Two integrable systems related to hyperbolic monopoles..

Hopf solitons on S3 and R3. (1999)
Journal Article
Ward, R. (1999). Hopf solitons on S3 and R3. Nonlinearity, 12(2), 241-246. https://doi.org/10.1088/0951-7715/12/2/005

The Skyrme–Faddeev system, a modified O(3) sigma model in three space dimensions, admits topological solitons with nonzero Hopf number. One may learn something about these solitons by considering the system on the 3-sphere of radius R. In particular,... Read More about Hopf solitons on S3 and R3..

Integrable systems: twistors, loop groups, and Riemann surfaces. (1999)
Book
Hitchin, N., Segal, G., & Ward, R. (1999). Integrable systems: twistors, loop groups, and Riemann surfaces. Oxford University Press

This textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are int... Read More about Integrable systems: twistors, loop groups, and Riemann surfaces..

Lax pairs for integrable lattice systems (1999)
Journal Article
Ward, R. (1999). Lax pairs for integrable lattice systems. Journal of Mathematical Physics, 40(1), 299-308. https://doi.org/10.1063/1.532772

This paper studies the structure of Lax pairs associated with integrable lattice systems (where space is a one-dimensional lattice, and time is continuous). It describes a procedure for generating examples of such systems, and emphasizes the features... Read More about Lax pairs for integrable lattice systems.

Integrable systems admitting topological solitons. (1998)
Journal Article
Ward, R., & Winn, A. (1998). Integrable systems admitting topological solitons. Journal of Physics A: Mathematical and Theoretical, 31(13), L261-L266. https://doi.org/10.1088/0305-4470/31/13/003

We discuss (1 + 1)-dimensional sigma models and Heisenberg models in which the target space has the topology of a cylinder. In these integrable systems, the solutions are classified by a winding number.

Twistors, geometry and integrable systems. (1998)
Book Chapter
Ward, R. (1998). Twistors, geometry and integrable systems. In S. A. Huggett, L. J. Mason, K. Tod, S. Tsou, & N. Woodhouse (Eds.), The Geometric Universe: Science, Geometry, and the Work of Roger Penrose (99-108). Oxford University Press

Bogomol'nyi Bounds for Two-Dimensional Lattice Systems. (1997)
Journal Article
Ward, R. (1997). Bogomol'nyi Bounds for Two-Dimensional Lattice Systems. Communications in Mathematical Physics, 184(2), 397-410. https://doi.org/10.1007/s002200050065

The O(3) sigma model and abelian Higgs model in two space dimensions admit topological (Bogomol'nyi) lower bounds on their energy. This paper proposes lattice versions of these systems which maintain the Bogomol'nyi bounds. One consequence is that in... Read More about Bogomol'nyi Bounds for Two-Dimensional Lattice Systems..

Stable topological Skyrmions on the 2D lattice. (1995)
Journal Article
Ward, R. (1995). Stable topological Skyrmions on the 2D lattice. Letters in Mathematical Physics, 35(4), 385-393. https://doi.org/10.1007/bf00750845

In the continuum O(3) sigma model in two spatial dimensions, there are topological solitons whose size can be stabilized by adding Skyrme and potential terms. This Letter describes a lattice version, namely a natural way of modifying the 2D Heisenber... Read More about Stable topological Skyrmions on the 2D lattice..

Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable system. (1995)
Journal Article
Ward, R. (1995). Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable system. Physics Letters A, 208(3), 203-208. https://doi.org/10.1016/0375-9601%2895%2900782-x

One usually expects localized solitons in an integrable system to interact trivially. There is an integrable (2+1)-dimensional chiral equation which admits multi-soliton solutions with trivial dynamics. This paper describes how to generate explicit s... Read More about Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable system..

Conserved quantities for integrable chiral equations in 2+1 dimensions. (1995)
Journal Article
Ioannidou, T., & Ward, R. (1995). Conserved quantities for integrable chiral equations in 2+1 dimensions. Physics Letters A, 208(3), 209-213. https://doi.org/10.1016/0375-9601%2895%2900781-w

The integrable (2+1)-dimensional chiral equations are related to the self-dual Yang-Mills equation. Previously known nonlocal conservation laws do not yield finite conserved charges, because the relevant spatial integrals diverge. We exhibit infinite... Read More about Conserved quantities for integrable chiral equations in 2+1 dimensions..

Discrete Toda field equations. (1995)
Journal Article
Ward, R. (1995). Discrete Toda field equations. Physics Letters A, 199(1-2), 45-48. https://doi.org/10.1016/0375-9601%2895%2900108-f

There are two-dimensional Toda field equations corresponding to each (finite or affine) Lie algebra. The question addressed in this note is whether there exist integrable discrete versions of these. It is shown that for certain algebras (such as An,... Read More about Discrete Toda field equations..

Numerical twistor procedure for solving a non-linear field equation. (1994)
Journal Article
Moorhouse, T., & Ward, R. (1994). Numerical twistor procedure for solving a non-linear field equation. Journal of Mathematical Physics, 35(12), 6489-6497. https://doi.org/10.1063/1.530686

This paper concentrates on an integrable SU(2) chiral equation in two space and one time dimensions, admitting soliton solutions. It is, in effect, a reduction of the self‐dual Yang–Mills equations in 2+2 dimensions, and can therefore be solved by tw... Read More about Numerical twistor procedure for solving a non-linear field equation..