This thesis is mainly concerned with the application of\r\ngroups of transformations to differential equations and in particular\r\nwith the connection between the group structure of a given equation\r\nand the existence of exact solutions and conservation laws. In this\r\nrespect the Lie-B\u00e4cklund groups of tangent transformations, particular\r\ncases of which are the Lie tangent and the Lie point groups,\r\nare extensively used.

\r\n\r\nIn Chapter I we first review the classical results of Lie,\r\nB\u00e4cklund and Bianchi as well as the more recent ones due mainly\r\nto Ovsjannikov. We then concentrate on the Lie-B\u00e4cklund groups\r\n(or more precisely on the corresponding Lie-B\u00e4cklund operators),\r\nas introduced by Ibragimov and Anderson, and prove some lemmas\r\nabout them which are useful for the following chapters. Finally\r\nwe introduce the concept of a conditionally admissible operator (as\r\nopposed to an admissible one) and show how this can be used to\r\ngenerate exact solutions.

\r\n\r\nIn Chapter II we establish the group nature of all separable\r\nsolutions and conserved quantities in classical mechanics by analyzing\r\nthe group structure of the Hamilton-Jacobi equation. It is\r\nshown that consideration of only Lie point groups is insufficient.\r\nFor this purpose a special type of Lie-B\u00e4cklund groups, those\r\nequivalent to Lie tangent groups, is used. It is also shown how\r\nthese generalized groups induce Lie point groups on Hamilton's\r\nequations. The generalization of the above results to any first\r\norder equation, where the dependent variable does not appear\r\nexplicitly, is obvious. In the second part of this chapter we\r\ninvestigate admissible operators (or equivalently constants of motion)\r\nof the Hamilton-Jacobi equation with polynornial dependence on the\r\nmomenta. The form of the most general constant of motion linear,\r\nquadratic and cubic in the momenta is explicitly found. Emphasis\r\nis given to the quadratic case, where the particular case of a fixed\r\n(say zero) energy state is also considered; it is shown that in the\r\nlatter case additional symmetries may appear. Finally, some\r\npotentials of physical interest admitting higher symmetries are considered.\r\nThese include potentials due to two centers and limiting\r\ncases thereof. The most general two-center potential admitting a \r\nquadratic constant of motion is obtained, as well as the corresponding\r\ninvariant. Also some new cubic invariants are found.

\r\n\r\nIn Chapter III we first establish the group nature of all\r\nseparable solutions of any linear, homogeneous equation. We then\r\nconcentrate on the Schrodinger equation and look for an algorithm\r\nwhich generates a quantum invariant from a classical one. The\r\nproblem of an isomorphism between functions in classical observables\r\nand quantum observables is studied concretely and constructively.\r\nFor functions at most quadratic in the momenta an isomorphism is\r\npossible which agrees with Weyl' s transform and which takes invariants\r\ninto invariants. It is not possible to extend the isomorphism\r\nindefinitely. The requirement that an invariant goes into an invariant \r\nmay necessitate variants of Weyl' s transform. This is illustrated\r\nfor the case of cubic invariants. Finally, the case of a\r\nspecific value of energy is considered; in this case Weyl's transform\r\ndoes not yield an isomorphism even for the quadratic case.\r\nHowever, for this case a correspondence mapping a classical\r\ninvariant to a quantum orie is explicitly found.

\r\n\r\nChapters IV and V are concerned with the general group\r\nstructure of evolution equations. In Chapter IV we establish a\r\none to one correspondence between admissible Lie-B\u00e4cklund\r\noperators of evolution equations (derivable from a variational\r\nprinciple) and conservation laws of these equations. This\r\ncorrespondence takes the form of a simple algorithm.

\r\n\r\nIn Chapter V we first establish the group nature of all\r\nB\u00e4cklund transformations (BT) by proving that any solution generated\r\nby a BT is invariant under the action of some conditionally\r\nadmissible operator. We then use an algorithm based on invariance\r\ncriteria to rederive many known BT and to derive some new\r\nones. Finally, we propose a generalization of BT which, among\r\nother advantages, clarifies the connection between the wave-train\r\nsolution and a BT in the sense that, a BT may be thought of as a\r\nvariation of parameters of some. special case of the wave-train\r\nsolution (usually the solitary wave one). Some open problems are\r\nindicated.

\r\n\r\nMost of the material of Chapters II and III is contained\r\nin [I], [II], [III] and [IV] and the first part of Chapter V\r\nin [V].

", "doi": "10.7907/A8EZ-HA03", "publication_date": "1979", "thesis_type": "phd", "thesis_year": "1979" } ]