This thesis is mainly concerned with the application ofgroups of transformations to differential equations and in particularwith the connection between the group structure of a given equationand the existence of exact solutions and conservation laws. In thisrespect the Lie-Bäcklund groups of tangent transformations, particularcases of which are the Lie tangent and the Lie point groups,are extensively used.
In Chapter I we first review the classical results of Lie,Bäcklund and Bianchi as well as the more recent ones due mainlyto Ovsjannikov. We then concentrate on the Lie-Bäcklund groups(or more precisely on the corresponding Lie-Bäcklund operators),as introduced by Ibragimov and Anderson, and prove some lemmasabout them which are useful for the following chapters. Finallywe introduce the concept of a conditionally admissible operator (asopposed to an admissible one) and show how this can be used togenerate exact solutions.
In Chapter II we establish the group nature of all separablesolutions and conserved quantities in classical mechanics by analyzingthe group structure of the Hamilton-Jacobi equation. It isshown that consideration of only Lie point groups is insufficient.For this purpose a special type of Lie-Bäcklund groups, thoseequivalent to Lie tangent groups, is used. It is also shown howthese generalized groups induce Lie point groups on Hamilton'sequations. The generalization of the above results to any firstorder equation, where the dependent variable does not appearexplicitly, is obvious. In the second part of this chapter weinvestigate admissible operators (or equivalently constants of motion)of the Hamilton-Jacobi equation with polynornial dependence on themomenta. The form of the most general constant of motion linear,quadratic and cubic in the momenta is explicitly found. Emphasisis given to the quadratic case, where the particular case of a fixed(say zero) energy state is also considered; it is shown that in thelatter case additional symmetries may appear. Finally, somepotentials of physical interest admitting higher symmetries are considered.These include potentials due to two centers and limitingcases thereof. The most general two-center potential admitting a quadratic constant of motion is obtained, as well as the correspondinginvariant. Also some new cubic invariants are found.
In Chapter III we first establish the group nature of allseparable solutions of any linear, homogeneous equation. We thenconcentrate on the Schrodinger equation and look for an algorithmwhich generates a quantum invariant from a classical one. Theproblem of an isomorphism between functions in classical observablesand quantum observables is studied concretely and constructively.For functions at most quadratic in the momenta an isomorphism ispossible which agrees with Weyl' s transform and which takes invariantsinto invariants. It is not possible to extend the isomorphismindefinitely. The requirement that an invariant goes into an invariant may necessitate variants of Weyl' s transform. This is illustratedfor the case of cubic invariants. Finally, the case of aspecific value of energy is considered; in this case Weyl's transformdoes not yield an isomorphism even for the quadratic case.However, for this case a correspondence mapping a classicalinvariant to a quantum orie is explicitly found.
Chapters IV and V are concerned with the general groupstructure of evolution equations. In Chapter IV we establish aone to one correspondence between admissible Lie-Bäcklundoperators of evolution equations (derivable from a variationalprinciple) and conservation laws of these equations. Thiscorrespondence takes the form of a simple algorithm.
In Chapter V we first establish the group nature of allBäcklund transformations (BT) by proving that any solution generatedby a BT is invariant under the action of some conditionallyadmissible operator. We then use an algorithm based on invariancecriteria to rederive many known BT and to derive some newones. Finally, we propose a generalization of BT which, amongother advantages, clarifies the connection between the wave-trainsolution and a BT in the sense that, a BT may be thought of as avariation of parameters of some. special case of the wave-trainsolution (usually the solitary wave one). Some open problems areindicated.
Most of the material of Chapters II and III is containedin [I], [II], [III] and [IV] and the first part of Chapter Vin [V].