Invariants, Lie-Bäcklund operators and Bäcklund transformations
<p>This thesis is mainly concerned with the application of groups of transformations to differential equations and in particular with the connection between the group structure of a given equation and the existence of exact solutions and conservation laws. In this respect the Lie-Bäcklund...
| Summary: | <p>This thesis is mainly concerned with the application of
groups of transformations to differential equations and in particular
with the connection between the group structure of a given equation
and the existence of exact solutions and conservation laws. In this
respect the Lie-Bäcklund groups of tangent transformations, particular
cases of which are the Lie tangent and the Lie point groups,
are extensively used.</p>
<p>In Chapter I we first review the classical results of Lie,
Bäcklund and Bianchi as well as the more recent ones due mainly
to 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 lemmas
about them which are useful for the following chapters. Finally
we introduce the concept of a conditionally admissible operator (as
opposed to an admissible one) and show how this can be used to
generate exact solutions.</p>
<p>In Chapter II we establish the group nature of all separable
solutions and conserved quantities in classical mechanics by analyzing
the group structure of the Hamilton-Jacobi equation. It is
shown that consideration of only Lie point groups is insufficient.
For this purpose a special type of Lie-Bäcklund groups, those
equivalent to Lie tangent groups, is used. It is also shown how
these generalized groups induce Lie point groups on Hamilton's
equations. The generalization of the above results to any first
order equation, where the dependent variable does not appear
explicitly, is obvious. In the second part of this chapter we
investigate admissible operators (or equivalently constants of motion)
of the Hamilton-Jacobi equation with polynornial dependence on the
momenta. The form of the most general constant of motion linear,
quadratic and cubic in the momenta is explicitly found. Emphasis
is given to the quadratic case, where the particular case of a fixed
(say zero) energy state is also considered; it is shown that in the
latter case additional symmetries may appear. Finally, some
potentials of physical interest admitting higher symmetries are considered.
These include potentials due to two centers and limiting
cases thereof. The most general two-center potential admitting a
quadratic constant of motion is obtained, as well as the corresponding
invariant. Also some new cubic invariants are found.</p>
<p>In Chapter III we first establish the group nature of all
separable solutions of any linear, homogeneous equation. We then
concentrate on the Schrodinger equation and look for an algorithm
which generates a quantum invariant from a classical one. The
problem of an isomorphism between functions in classical observables
and quantum observables is studied concretely and constructively.
For functions at most quadratic in the momenta an isomorphism is
possible which agrees with Weyl' s transform and which takes invariants
into invariants. It is not possible to extend the isomorphism
indefinitely. The requirement that an invariant goes into an invariant
may necessitate variants of Weyl' s transform. This is illustrated
for the case of cubic invariants. Finally, the case of a
specific value of energy is considered; in this case Weyl's transform
does not yield an isomorphism even for the quadratic case.
However, for this case a correspondence mapping a classical
invariant to a quantum orie is explicitly found.</p>
<p>Chapters IV and V are concerned with the general group
structure of evolution equations. In Chapter IV we establish a
one to one correspondence between admissible Lie-Bäcklund
operators of evolution equations (derivable from a variational
principle) and conservation laws of these equations. This
correspondence takes the form of a simple algorithm.</p>
<p>In Chapter V we first establish the group nature of all
Bäcklund transformations (BT) by proving that any solution generated
by a BT is invariant under the action of some conditionally
admissible operator. We then use an algorithm based on invariance
criteria to rederive many known BT and to derive some new
ones. Finally, we propose a generalization of BT which, among
other advantages, clarifies the connection between the wave-train
solution and a BT in the sense that, a BT may be thought of as a
variation of parameters of some. special case of the wave-train
solution (usually the solitary wave one). Some open problems are
indicated.</p>
<p>Most of the material of Chapters II and III is contained
in [I], [II], [III] and [IV] and the first part of Chapter V
in [V].</p> |
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