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Toda Lattices, Cosymplectic Manifolds, Backlund Transformations and Kinks, part B (Interdisciplinary Mathematics Series, Vol 18): Robert Hermann

The volume contains material from my research work and lectures of the last six months, and develops several new ideas and conceps. Chapters I and X are the most important. The former describes what I believe is a major new idea - the description of the ``scattering data" associated with a Sturm-Liouville equation in terms of its Riccati equation and the geometry of the compact homogeneous space of the Lie group SL(2,R). I believe the introduction of this more ``geometric" point of view wil facilitate genralizations, with other Lie groups replacing SL(2,R). Similarly, in Chapter X I prepare the way for the generalization of the ``soliton" concept to other Lie groups and homogeneous spaces, as well as make what I believe is at least a first approximation of a guess as to what ``solitons" will be for field theories in two or more space dimensions. (The problem of extending the ideas to higher dimensions is the most pressing one for the ultimate physical applications!).

Contents THE GEOMETRY OF SCATTERING THEORY AND BIANCHI 1 scattering theory, cross-ratio, Riccati equation Asymptotic Behavior of the Superposition Formula 10 Riccati equation, DYNAMICAL SYSTEMS, Backlund transformations THE GENERAL GEOMETRIC THEORY OF CONNECTIONS 21 Lie algebra, Korteweg-de Vries equation, vector fields vii 79 THE RELATION BETWEEN BACKLUND TRANSFORMATIONS 43 Sine-Gordon, one-forms, Wahlquist THE BACKLUND TOWER 53 one-forms, partial differential equation, two-form THE ALGEBRAIC PROLONGATION STRUCTURE FOR KORTEWEGDE VRIES 63 Lie algebra, Korteweg-de Vries equation, ALGEBRA STRUCTURES NONLINEAR WAVE EQUATIONS DEFINED BY GENERALIZED 71 Sine-Gordon equation, CONSERVATION LAWS, B])dx A dt Chaptei 79 Lie group, integrability conditions, scalar A SPECULATIVE LIETHEORETIC SETTING FOR DIFFERENTIAL 89 CONSERVATION LAWS, soliton, Lie subgroup THE PROLONGATION EQUATIONS OF NONLINEAR WAVE THEORY 103 Toda lattice, Sanuki, Ansatz THE GROUPTHEORETIC AND TOPOLOGICAL STRUCTURE OF ONE 111 SOLITONS, one-parameter subgroup, GROUP THEORY The Differential Equation for the Kortewegde Vries 117 FUNDAMENTAL GROUP, homogeneous spaces, kink l0 Examples of Compact Homogeneous Spaces GG+ KKQ 128 Killing form, homotopy group, eigenvalues l2 Speculations about Elementary Particle Physics 134 DIFFERENTIAL EQUATIONS WHICH ADMIT SOLITON SOLUTIONS 139 Homogeneous Spaces, submanifold, Toda lattice

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