This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

/pdf/backlund-and-darboux-transformations-geometry-and-modern-4k80wq4g1i.pdf

Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control theoretical applications. The basic methodology is that of geometric mechanics applied to the formulation of Lagrange d'Alembert, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a ``body reference frame'' relates part of the momentum equation to the components of the Euler-Poincar\'{e} equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory. September 1994 Revised, March 1995 Revised, June 1995

/pdf/nonholonomic-mechanical-systems-with-symmetry-s3tqskxtsb.pdf

Nonholonomic Mechanical Systems with Symmetry

A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three-dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three-leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three-dimensional integrable systems and to the quantum context are also discussed.

Classification of integrable equations on quad-graphs. The consistency approach

The main purpos e of this chapter is to present a direct and systematic way of finding exact solutions and Backlund transformations of a certain class of nonlinear evolution equations. The nonlinear evolution equations are transformed, by changing the dependent variable(s), into bilinear differential equations of the following special form 
 
$$ F\left( {\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial {t^1}}},\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial {x^1}}}} \right)f(t,x)f({t^1},{x^1}){|_{t = {t^1},x = {x^1}}} = 0 $$ 
 
, which we solve exactly using a kind of perturbational approach.

Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告)

Variational iteration method for solving Burger's and coupled Burger's equations

Two novel classes of integrable surfaces (“generalized Weingarten surfaces of Class 1 and 2”) which may be defined in a coordinate-independent manner or, equivalently, via an extension of the classical Lelieuvre formulas are introduced. The classes 1 and 2 generalize Bianchi surfaces and surfaces of harmonic inverse mean curvature respectively and turn out to be governed by analogs of the Ernst–Weyl equation descriptive of gravitational and neutrino fields in general relativity. The latter connection is exploited to derive Laplace–Darboux-type transformations for Class 2 via the associated Loewner–Konopelchenko–Rogers representation. The Laplace–Darboux-type transformations are shown to have remarkable geometric properties in terms of sphere congruences. Connections with the Pohlmeyer–Lund–Regge vortex model and an inhomogeneous Heisenberg spin equation are recorded. A Backlund transformation for a particular Painleve III equation is derived in a purely geometric manner and a geometric interpretation of a...

On Laplace–Darboux-type sequences of generalized Weingarten surfaces

A form of the linear equations governing hyperbolic surfaces in R 3 which may be regarded as the Gaus equations in centro-affine geometry is derived. Based on this formulation, a novel class of integrable surfaces which generalize affine spheres is obtained. The Tzitzeica transformation is used to generate their integrable discrete counterparts.

Hyperbolic surfaces in centro-affine geometry: Integrability and discretization

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.

Hidden integrability in ideal magnetohydrodynamics: The Pohlmeyer–Lund–Regge model

On a maximum principle for minimal surfaces and their integrable discrete counterparts

Backlund and Moutard transformations are applied to seed classical isothermic surfaces to generate 'solitonic' isothermic surfaces such as Dupin cyclides. Corresponding localized solutions of the zoomeron equation are shown to be related to the dromion solutions of the Davey-Stewartson III equation via a simple Lie point symmetry.

Wolfgang K. Schief

Papers

On Laplace–Darboux-type sequences of generalized Weingarten surfaces

Hyperbolic surfaces in centro-affine geometry: Integrability and discretization

Hidden integrability in ideal magnetohydrodynamics: The Pohlmeyer–Lund–Regge model

On a maximum principle for minimal surfaces and their integrable discrete counterparts

Isothermic Surfaces Generated via Bäcklund and Moutard Transformations: Boomeron and Zoomeron Connections