Showing papers by "Wolfgang K. Schief published in 2003"
TL;DR: In this article, the authors derived mobius invariant versions of the discrete Darboux, KP, BKP and CKP equations by imposing elementary geometric constraints on an (irregular) lattice in a three-dimensional Euclidean space.
Abstract: Mobius invariant versions of the discrete Darboux, KP, BKP and CKP equations are derived by imposing elementary geometric constraints on an (irregular) lattice in a three-dimensional Euclidean space. Each case is represented by a fundamental theorem of plane geometry. In particular, classical theorems due to Menelaus and Carnot are employed. An interpretation of the discrete CKP equation as a permutability theorem is also provided.
62 citations
TL;DR: In this paper, a novel class of discrete integrable surfaces, including discrete analogues of classical surfaces such as isothermic, linear Weingarten, Guichard and Petot surfaces, is presented.
Abstract: A novel class of discrete integrable surfaces is recorded This class of discrete O surfaces is shown to include discrete analogues of classical surfaces such as isothermic, ‘linear’ Weingarten, Guichard and Petot surfaces Moreover, natural discrete analogues of the Gaussian and mean curvatures for surfaces parametrized in terms of curvature coordinates are used to define surfaces of constant discrete Gaussian and mean curvatures and discrete minimal surfaces Remarkably, these turn out to be prototypical examples of discrete O surfaces It is demonstrated that the construction of a Backlund transformation for discrete O surfaces leads in a natural manner to an associated parameter–dependent linear representation Canonical discretizations of the classical pseudosphere and breather pseudospherical surfaces are generated Connections with pioneering work by Bobenko and Pinkall are established
50 citations
TL;DR: In this article, integrable maps defined on face-centred (fcc) lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete KP and BKP equations and the integrably discrete Darboux system governing conjugate lattices.
Abstract: Geometric and algebraic aspects of multi-ratios M2N are investigated in detail. Connections with Menelaus' theorem, Clifford configurations and Maxwell's reciprocal quadrangles are utilized to associate the multi-ratios M4, M8 and M8 with tetrahedra, octahedra and cubo-octahedra respectively. Integrable maps defined on face-centred (fcc) lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete KP and BKP equations and the integrable discrete Darboux system governing conjugate lattices. An interpretation in terms of integrable irregular lattices of slopes on the plane is also given.
33 citations
TL;DR: In this article, it was shown that the classical magnetohydrostatic equations of an infinitely conducting fluid reduce to the integrable potential Heisenberg spin equation subject to a Jacobian condition if the magnitude of the magnetic field is constant along individual magnetic field lines.
Abstract: It is shown that the classical magnetohydrostatic equations of an infinitely conducting fluid reduce to the integrable potential Heisenberg spin equation subject to a Jacobian condition if the magnitude of the magnetic field is constant along individual magnetic field lines. Any solution of the constrained potential Heisenberg spin equation gives rise to a multiplicity of magnetohydrostatic equilibria which share the magnetic field line geometry. The multiplicity of equilibria is reflected by the local arbitrariness of the total pressure profile. A connection with the classical Da Rios equations is exploited to establish the existence of associated helically and rotationally symmetric equilibria. As an illustration, Palumbo's ‘unique’ toroidal isodynamic equilibrium is retrieved.
31 citations
TL;DR: In this paper, it was shown that the equilibrium equations of ideal magnetohydrodynamics reduce to the integrable Pohlmeyer-Lund-Regge model subject to a volume-preserving constraint if the Maxwellian surfaces are assumed to coincide with the constant total pressure surfaces.
Abstract: It is shown that the equilibrium equations of ideal magnetohydrodynamics reduce to the integrable Pohlmeyer–Lund–Regge model subject to a volume-preserving constraint if the Maxwellian surfaces are assumed to coincide with the constant total pressure surfaces. It is demonstrated that any solution of the constrained Pohlmeyer–Lund–Regge model gives rise to a multiplicity of solutions of the magnetohydrodynamic system which share the streamline and magnetic field line geometry. Explicit solutions given in terms of elliptic functions and integrals are constructed.
28 citations
TL;DR: In this article, the integrability of Bertrand curves and their geodesic embedding in surfaces is discussed in the context of modern soliton theory, and the existence of parallel Razzaboni surfaces which constitute the surface analogues of the classical offset Bertrand mates is recorded.
27 citations
TL;DR: In this paper, a master 2+1-dimensional soliton system was analyzed and the classical symmetries were shown to constitute an infinite dimensional Kac-Moody-Virasoro algebra.
Abstract: A symmetry analysis is conducted for a master 2+1-dimensional soliton system. The classical symmetries are shown to constitute an infinite dimensional Kac–Moody–Virasoro algebra. Finite symmetry group transformations are then used to construct localized excitations of the system.
24 citations
TL;DR: In this article, hidden integrable structure is revealed in diverse areas of nonlinear continuum mechanics through natural geometric constraints, and the nonlinear equations that describe solitonic behavior in physical systems have, to date, typically been derived by approximation or expansion methods.
Abstract: The nonlinear equations that describe solitonic behavior in physical systems have, to-date, 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.
19 citations
TL;DR: In this article, it was shown that a well-known system of classical shell theory describing membranes in equilibrium is integrable, and the membranes are shown to have geometries within the integrability class of so-called O surfaces, including minimal, constant mean curvature, constant Gaussian curvature and linear Weingarten surfaces.
Abstract: It is established that a well–known system of classical shell theory descriptive of membranes in equilibrium is, in fact, integrable. The membranes are shown to have geometries within the integrable class of so–called O surfaces. The membrane O surfaces include inter alia minimal, constant mean curvature, constant Gaussian curvature and, more generally, linear Weingarten surfaces, as well as canal surfaces and Dupin cyclides.
18 citations
TL;DR: In this article, a geometric formulation for the kinematic conditions attendant upon the motion of an ideal fiber-reinforced fluid is presented. But it is only for the case of planar motion, and the conditions admit a reduction to a solitonic system related to the classical sine-Gordon equation.
Abstract: A geometric formulation previously adopted in hydrodynamics and soliton theory is used here to investigate the kinematic conditions attendant upon the motion of an ideal fibre-reinforced fluid. Conditions are established for the existence of multiple-fibre configurations. Kinematically admissible spatial motions are obtained in which the fibres are geodesic windings on nested toroidal surfaces. In the case of purely planar motion, it is shown that the kinematic relations reduce to a third-order nonlinear equation. Remarkably, this admits a reduction to a solitonic system which is related to the classical sine-Gordon equation. The kinematic conditions in this case possess a duality property.
9 citations
TL;DR: In this paper, the kinematic constraints on the steady planar motion of an ideal fiber-reinforced fluid can be consolidated in a single third-order nonlinear equation, which admits a solitonic reduction related to the classical sine-Gordon equation.
Abstract: We establish that the kinematic constraints on the steady planar motion of an ideal fiber-reinforced fluid can be consolidated in a single third-order nonlinear equation. Remarkably, this equation admits a solitonic reduction related to the classical sine-Gordon equation. The kinematic conditions in this case admit a novel duality property and a Backlund transformation.
TL;DR: In this paper, exact representations for the stress distribution evolution in model elastic materials are obtained corresponding to classical Beltrami and Dini surfaces as well as a two-soliton pseudospherical surface generated via the classical Backlund transformation.
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TL;DR: The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated in this article, where connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded.
Abstract: The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.