Lie-Bäcklund Transformations in Applications

Description

This title presents an introduction to the classical treatment of Backlund and general surface transformations; and includes detailed and accessible techniques for constructing both groups of tranformations which will be of great value to the scientist and engineer in the analysis of mathematical models of physical phenomena. Classical and recent examples of Backlund transformations as applied to geometry, nonlinear optics, turbulence models, nonlinear waves and quantum mechanics are given. The authors discuss applications of Lie-Backlund transformations in mechanics, quantum mechanics, gas dynamics, hydrodynamics, and relativity.

MR0520395

This book provides an introduction to the classical theory of Lie tangent transformations and Bäcklund transformations and presents a recent generalisation of them, termed Lie-Bäcklund tangent transformations. The general theory is only briefly outlined but many interesting examples and applications are discussed in detail. Chapter I is devoted to the history of Lie tangent transformations operating in the space of variables x 1,⋯,x n , u ,u 1,⋯,u n and preserving the equation d u −∑ u j d x j =0. It is shown that there are only two possible ways for further development. The first is the Lie-Bäcklund tangent transformation (a nonclassical notion) which acts in the infinite-dimensional space x 1,⋯,x n ,u 1,⋯,u m ,u 11,⋯,u α i 1⋯ i s ,⋯ (this is not made precise in the book) and preserves the infinite system d u α −∑ u α j d x j =d u α i −∑ u α i j d x j =⋯=0. The second method (classical) is the theory of Bäcklund transformations, operating in the simplest (but most common) case in the space of surface elements x ,y ,z ,p ,q. It is not a one-to-one transformation but behaves well on the system of all solutions of a certain related partial differential equation of second order which are transformed into solutions of the same equation or into solutions of another one. This property is illustrated on the Lie-Bianchi transformation of surfaces of constant curvature, on the sine-Gordon equation z x y =sin z , the Kortewegde Vries (KdV) equation u y +6 u u x +u x x x =0 and on the modified KdV equation u y +6 u 2 u x x +u x x x =0, which are invariant. Also, the Liouville equation u x y =e u and the KdV equation can be transformed into the wave equation u x y =0 and the modified KdV equation, respectively. Chapter II deals with the general theory and repeats the preceding themes using the technique of infinitesimal transformations. Chapter III deals with the invariance of a system of differential equations with respect to the group of Lie-Bäcklund transformations, with conservation laws for variational problems, and contains many illustrative examples from mechanics. There is also a thorough analysis of the equation − F x F x x −2 F x t +F y y =0 of unsteady gas motion, the Dirac equation ∑ c k ∂ u /∂ x k +m u =0, the equations for ideal polytropic gas flow and for incompressible fluid flow, the Schrödinger equation and the dynamical system d u α /d x =F(u α), d v α /d x =G(u α). In the shortest chapter, Chapter IV, the authors describe applications to nonlinear optics, lattices, diffusion and nonlinear waves. Reviewed by J. Chrastina