W. K. Schief

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

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are BäcklundDarboux 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 Gauss-MainardiCodazzi equations for various types of surfaces that admit invariance under Bäcklund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physicists.

Superposition principles associated with the Moutard transformation: an integrable discretization of a (2+1)–dimensional sine–Gordon system

Superposition principles, both linear and nonlinear, associated with the Moutard transformation are found. On suitable reinterpretation, these constitute an integrable discrete nonlinear system and its associated linear system. Further, it is shown that, in a particular form, this system is an integrable discretization of a (2+1)–dimensional sine–Gordon system. Solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the Moutard transformation. Included in these solutions are discrete analogues of the kink solutions of the continuous system.

Three–dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality

It is shown that the discrete Darboux system, descriptive of conjugate lattices in Euclidean spaces, admits constraints on the (adjoint) eigenfunctions which may be interpreted as discrete orthogonality conditions on the lattices. Thus, it turns out that the elementary quadrilaterals of orthogonal lattices are cyclic. Orthogonal lattices on lines, planes and spheres are discussed and the underlying integrable systems in one, two and three dimensions are derived explicitly. A discrete analogue of Bianchis Ribaucour transformation is set down and particular orthogonal lattices are given. As a by–product, discrete Dini surfaces are obtained.

Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces

The purely binormal motion of curves of constant curvature or torsion, respectively, is shown to lead to integrable extensions of the Dym and classical sine–Gordon equations. In the case of the extended Dym equation, a reciprocal invariance is used to establish the existence of novel dual–soliton surfaces associated with a given soliton surface. A cc–ideal formulation is adduced to obtain a matrix Darboux invariance for the extended Dym and reciprocally linked m2KdV equations. A Bäcklund transformation with a classical constant–length property is thereby constructed which allows the generation of associated soliton surfaces. Analogues of both Bäcklunds and Bianchis classical transformations are derived for the extended sine–Gordon system.

SQUARED EIGENFUNCTIONS OF THE (MODIFIED) KP HIERARCHY AND SCATTERING PROBLEMS OF LOEWNER TYPE

It is shown that products of eigenfunctions and (integrated) adjoint eigenfunctions associated with the (modified) Kadomtsev-Petviashvili (KP) hierarchy form generators of a symmetry transformation. Linear integro-differential representations for these symmetries are found. For special cases the corresponding nonlinear equations are the compatibility conditions of linear scattering problems of Loewner type. The examples include the 2+1-dimensional sine-Gordon equation with space variables occuring on an equal footing introduced recently by Konopelchenko and Rogers. This equation represents a special squared eigenfunction symmetry of the Ishimori hierarchy.

Isothermic Surfaces in Spaces of Arbitrary Dimension: Integrability, Discretization, and Bäcklund Transformations—A Discrete Calapso Equation

A vector analog of the classical Calapso equation governing isothermic surfaces in R n+2 is introduced. It is shown that this vector Calapso system admits a (nonlocal) scalar Lax pair based on the classical Moutard equation. The analog of Darbouxs Backlund transformation for isothermic surfaces in R 3 is derived in a systematic manner and shown that it may be formulated in terms of the classical Moutard transformation acting on the scalar Lax pair. A permutability theorem for isothermic surfaces is set down that manifests itself in an explicit superposition principle for the vector Calapso system. This superposition principle in vectorial form is shown to constitute an integrable discretization of the vector Calapso system and, therefore, defines discrete isothermic surfaces in R n+2 , The discrete Calapso equation is related to the discrete Korteweg-de Vries equation and discrete holomorphic functions. A matrix Lax pair based on Clifford algebras and a scalar Lax pair are derived for the discrete Calapso equation. A discrete Moutard-type transformation for the discrete Calapso equation is obtained, and it is shown that the discrete Calapso equation may be specialized to an integrable discrete version of the O(n +2) nonlinear σ-model.

The resonant nonlinear Schrödinger equation in cold plasma physics. Application of Bäcklund-Darboux transformations and superposition principles

A system of nonlinear equations governing the transmission of uni-axial waves in a cold collisionless plasma subject to a transverse magnetic field is reduced to the recently proposed resonant nonlinear Schrodinger (RNLS) equation. This integrable variant of the standard nonlinear Schrodinger equation admits novel nonlinear superposition principles associated with Backlund-Darboux transformations. These are used here, in particular, to construct analytic descriptions of the interaction of solitonic magnetoaeoustie waves propagating through the plasma.

On the unification of classical and novel integrable surfaces. II. Difference geometry

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 Bäcklund 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.

Lattice Geometry of the Discrete Darboux, KP, BKP and CKP Equations. Menelaus' and Carnot's Theorems

Abstract Möbius 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.

Ermakov systems of arbitrary order and dimension: structure and linearization

Ermakov systems of arbitrary order and dimension are constructed. These inherit an underlying linear structure based on that recently established for the classical Ermakov system. As an application, alignment of a (2 + 1)-dimensional Ermakov and integrable Ernst system is shown to produce a novel integrable hybrid of a (2 + 1)-dimensional sinh - Gordon system and of a conventional Ermakov system.