Volume 44, Issue 8, August 2003
 SPECIAL ISSUE: INTEGRABILITY, TOPOLOGICAL SOLITONS AND BEYOND

 CLASSICAL INTEGRABLE EQUATIONS

Integrable systems and reductions of the selfdual Yang–Mills equations
View Description Hide DescriptionMany integrable equations are known to be reductions of the selfdual Yang–Mills equations. This article discusses some of the well known reductions including the standard soliton equations, the classical Painlevé equations and integrable generalizations of the Darboux–Halphen system and Chazy equations. The Chazy equation, first derived in 1909, is shown to correspond to the equations studied independently by Ramanujan in 1916.

Topological phenomena in the real periodic sineGordon theory
View Description Hide DescriptionThe set of real finitegap sineGordon solutions corresponding to a fixed spectral curve consists of several connected components. A simple explicit description of these components obtained by the authors recently is used to study the consequences of this property. In particular this description allows to calculate the topological charge of solutions (the averaging of the xderivative of the potential) and to show that the averaging of other standard conservation laws is the same for all components.

Generation of asymptotic solitons of the nonlinear Schrödinger equation by boundary data
View Description Hide DescriptionThis article is about the focusing nonlinear Schrödinger equation on the halfline. The initial function vanishes at infinity while boundary data are local perturbations of periodic or quasiperiodic (finitegap) functions. We study the corresponding scattering problem for the Zakharov–Shabat compatible differential equations, the representation of the solution of the nonlinear Schrödinger equation in the quarter of the plane through functions, which satisfy Marchenko integral equations. We use this formalism to investigate the asymptotic behavior of the solution for large time. We prove that under certain conditions a periodic (quasiperiodic) behavior at infinity of boundary data generates an unbounded train of asymptotic solitons running away from the boundary. The asymptotics of the solution shows that boundary data with periodic behavior as time tends to infinity generates a train of such asymptotic solitons even in the case when the initial function is identically zero.

Lax equations scattering and KdV
View Description Hide DescriptionThe study of the Korteveg–de Vries (KdV) equation is considered as a special chapter of potential scattering where the dynamic scattering equation is a set of coupled “Lax” equations. With this approach, all points of view and all tools of potential scattering have their counterpart in the standard inverse scattering transform, which appears as a straightforward consequence. If the approach is transposed to the quarterplane problem, it shows a generalization to KdV of the solutions obtained by Fokas in the linearized KdV problem, but unfortunately the last step is iterative and complicated. The method can also be used to study NLS.

The Davey–Stewartson I equation on the quarter plane with homogeneous Dirichlet boundary conditions
View Description Hide DescriptionDromions are exponentially localized coherent structures supported by nonlinear integrable evolution equations in two spatial dimensions. In the study of initialvalue problems on the plane, such solutions occur only if one imposes nontrivial boundary conditions at infinity, a situation of dubious physical significance. However, it is established here that dromions appear naturally in the study of boundaryvalue problems. In particular, it is shown that the long time asymptotics of the solution of the Davey–Stewartson I equation in the quarter plane with arbitrary initial conditions and with zero Dirichlet boundary conditions is dominated by dromions. The case of nonzero Dirichlet boundary conditions is also discussed.

The ncomponent KP hierarchy and representation theory
View Description Hide DescriptionIt is the aim of the present article to give all formulations of the ncomponent KP hierarchy and clarify connections between them. The generalization to the ncomponent KP hierarchy is important because it contains many of the most popular systems of soliton equations, like the Davey–Stewartson system (for the twodimensional Toda lattice (for the nwave system (for and the Darboux–Egoroff system. It also allows us to construct natural generalizations to the Davey–Stewartson and Toda lattice systems. Of course, the inclusion of all these systems in the ncomponent KP hierarchy allows us to construct their solutions by making use of vertex operators.

Additional symmetries and solutions of the dispersionless KP hierarchy
View Description Hide DescriptionThe dispersionless KP hierarchy is considered from the point of view of the twistor formalism. A set of explicit additional symmetries is characterized and its action on the solutions of the twistor equations is studied. A method for dealing with the twistor equations by taking advantage of hodograph type equations is proposed. This method is applied for determining the orbits of solutions satisfying reduction constraints of Gelfand–Dikii type under the action of additional symmetries.

Extended resolvent and inverse scattering with an application to KPI
View Description Hide DescriptionWe present in detail an extended resolvent approach for investigating linear problems associated to dimensional integrable equations. Our presentation is based as an example on the nonstationary Schrödinger equation with potential being a perturbation of the onesoliton potential by means of a decaying twodimensional function. Modification of the inverse scatteringtheory as well as properties of the Jost solutions and spectral data as follows from the resolvent approach are given.
 GEOMETRY AND INTEGRABILITY

Novel integrable reductions in nonlinear continuum mechanics via geometric constraints
View Description Hide DescriptionThe nonlinear equations that describe solitonic behavior in physical systems have, todate, typically been derived by approximation or expansion methods. Here, by contrast, hidden integrable structure is revealed in diverse areas of nonlinear continuum mechanics through natural geometric constraints.

harmonic maps and the Weierstrass problem
View Description Hide DescriptionA Weierstrasstype system of equations corresponding to the harmonic maps is presented. The system constitutes a further generalization of our previous construction [J. Math. Phys. 44, 328 (2003)]. It consists of four first order equations for three complex functions which are shown to be equivalent to the harmonic maps. When the harmonic maps are holomorphic (or antiholomorphic) one of the functions vanishes and the system reduces to the previously given generalization of the Weierstrass problem. We also discuss a possible interpretation of our results and show that in our new case the induced metric is proportional to the total energy density of the map and not only to its holomorphic part, as was the case in the previous generalizations.

Painlevé analysis of the Ricciflat ordinary differential equations associated with Aloff–Wallach spaces and U(1)bundles over Fano products
View Description Hide DescriptionWe apply techniques of Painlevé–Kowalewski analysis to certain ODE reductions of the Ricciflat equations. We particularly focus on two examples when the hypersurface is an Aloff–Wallach space or a circle bundle over a Fano product.

Progress in relativistic gravitational theory using the inverse scattering method
View Description Hide DescriptionThe increasing interest in compact astrophysical objects (neutron stars,binaries, galactic black holes) has stimulated the search for rigorous methods, which allow a systematic general relativistic description of such objects. This article is meant to demonstrate the use of the inverse scattering method, which allows, in particular cases, the treatment of rotating body problems. The idea is to replace the investigation of the matter region of a rotating body by the formulation of boundary values along the surface of the body. In this way we construct solutions describing rotating black holes and disks of dust (“galaxies”). Physical properties of the solutions and consequences of the approach are discussed. Among other things, the balance problem for two black holes can be tackled.

Twistor theory of hyperKähler metrics with hidden symmetries
View Description Hide DescriptionWe review the hierarchy for the hyperKähler equations and define a notion of symmetry for solutions of this hierarchy. A fourdimensional hyperKähler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyperKähler metric admits a hidden symmetry if it admits a certain Killing spinor. We show that if the hidden symmetry is triholomorphic, then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a “twistor group” action as introduced by Bielawski [Twistor Quotients of HyperKähler Manifolds (World Scientific, River Edge, NJ, 2001)]. This leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalized Legendre transform for the hyperKähler metric itself given by Ivanov and Rocek [Commun. Math. Phys. 182, 291 (1996)]. We show that the ALE spaces are examples of hyperKähler metrics admitting three triholomorphic Killing spinors. These metrics are in this sense analogous to the “finite gap” solutions in soliton theory. Finally we extend the concept of a hierarchy from that of our earlier work [Commun. Math. Phys. 213, 641 (2000)] for the fourdimensional hyperKähler equations to a generalization of the conformal antiselfduality equations and briefly discuss hidden symmetries for these equations.

Hexagonal circle patterns with constant intersection angles and discrete Painlevé and Riccati equations
View Description Hide DescriptionHexagonal circle patterns with constant intersection angles mimicking holomorphic maps and are studied. It is shown that the corresponding circle patterns are immersed and described by special separatrix solutions of discrete Painlevé and Riccati equations. The general solution of the Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solutions, as well as of the discrete and are established.
 TOPOLOGICAL SOLITONS

The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps
View Description Hide DescriptionThe most fruitful approach to studying low energy soliton dynamics in field theories of Bogomol’nyi type is the geodesic approximation of Manton. In the case of vortices and monopoles, Stuart has obtained rigorous estimates of the errors in this approximation, and hence proved that it is valid in the low speed regime. His method employs energy estimates which rely on a key coercivity property of the Hessian of the energy functional of the theory under consideration. In this article we prove an analogous coercivity property for the Hessian of the energy functional of a general sigma model with compact Kähler domain and target. We go on to prove a continuity property for our result, and show that, for the model on the Hessian fails to be globally coercive in the degree 1 sector. We present numerical evidence which suggests that the Hessian is globally coercive in a certain equivariance class of the degree sector for We also prove that, within the geodesic approximation, a single lump moving on does not generically travel on a great circle.

The dynamics of vortices on near the Bradlow limit
View Description Hide DescriptionThe explicit solutions of the Bogomolny equations for vortices on a sphere of radius are not known. In particular, this has prevented the use of the geodesic approximation to describe the low energy vortex dynamics. In this article we introduce an approximate general solution of the equations, valid for which has many properties of the true solutions, including the same moduli space Within the framework of the geodesic approximation, the metric on the moduli space is then computed to be proportional to the Fubini–Study metric, which leads to a complete description of the particle dynamics.

The moduli space of noncommutative vortices
View Description Hide DescriptionThe Abelian Higgs model on the noncommutative plane admits both BPS vortices and nonBPS fluxons. After reviewing the properties of these solitons, we discuss several new aspects of the former. We solve the Bogomoln’yi equations perturbatively, to all orders in the inverse noncommutivity parameter, and show that the metric on the moduli space of vortices reduces to the computation of the trace of a dimensional matrix. In the limit of large noncommutivity, we present an explicit expression for this metric.

A note on monopole moduli spaces
View Description Hide DescriptionWe discuss the structure of the framed moduli space of Bogomolny monopoles for arbitrary symmetry breaking and extend the definition of its stratification to the case of arbitrary compact Lie groups. We show that each stratum is a union of submanifolds for which we conjecture that the natural metric is hyperKähler. The dimensions of the strata and of these submanifolds are calculated, and it is found that for the latter, the dimension is always a multiple of four.

Polyhedral scattering of fundamental monopoles
View Description Hide DescriptionThe dynamics of slowly moving fundamental monopoles in the BPS Yang–Mills–Higgs theory can be approximated by geodesic motion on the dimensional hyperkähler Lee–Weinberg–Yi manifold. In this article we apply a variational method to construct some scaling geodesics on this manifold. These geodesics describe the scattering of monopoles which lie on the vertices of a bouncing polyhedron; the polyhedron contracts from infinity to a point, representing the spherically symmetric monopole, and then expands back out to infinity. For different monopole masses the solutions generalize to form bouncing nested polyhedra. The relevance of these results to the dynamics of well separated SU(2) monopoles is also discussed.

Icosahedral Skyrmions
View Description Hide DescriptionIn this article we aim to determine the baryon numbers at which the minimal energy Skyrmion has icosahedral symmetry. By comparing polyhedra which arise as minimal energy Skyrmions with the dual of polyhedra that minimize the energy of Coulomb charges on a sphere, we are led to conjecture a sequence of magic baryon numbers, at which the minimal energy Skyrmion has icosahedral symmetry and unusually low energy. We present evidence for this conjecture by applying a simulated annealing algorithm to compute energy minimizing rational maps for all degrees up to 40. Further evidence is provided by the explicit construction of icosahedrally symmetric rational maps of degrees 37, 47, 67 and 97. To calculate these maps we introduce two new methods for computing rational maps with Platonic symmetries.