This thesis is
concerned about various aspects of 1 + 1
dimensional non-linear evolution equations. In
particular, it is devoted to the study of
those properties which can be obtained by the
application of non-linear transformations,
such as Cole-Hopf, Miura and reciprocal-type
transformations. The concept of Backlund Chart
is introduced to depict succinctly the various
relationships interconnecting different
non-linear systems. An account of the Painleve
Test, an integrability test for non-linear
partial differential equations, is given. The
Painleve Analysis indicates how to relate to
non-linear integrable systems equations,
termed "singularity manifold" equations, the
latter exhibit a particular structure which
plays a key role in the present investigation.
An overview on reciprocal-type transformations
follows, especially in connection with their
importance in linking non-linear evolution
equations. Then, a Backlund Chart which
comprises the Caudrey-Dodd-Gibbon (GDG) and
Kaup-Kupershmidt (KK) is constructed. It
provides an explicit link between such
equations and their, respective singularity
manifold equations. New hierarchies of
integrable non-linear evolution equations are
obtained via reciprocal-type transformations.
They exhibit a novel invariance and have base
member a Kawamoto-type equation. Furthermore,
the spatial part of new generic auto-Backlund
transformations, for both the CDG and KK
hierarchies are constructed. The symmetry
structure of a Kawamoto equation is
subsequently studied. The Hamiltonian and
bi-Hamiltonian formulation of the Kawamoto
equation, as well as its hereditary recursion
operator are obtained via the links in the
Backlund Chart. Subsequently, an analog study
is developed for non-linear systems related to
the Korteweg-deVries (KdV) equation. An
extensive Backlund Chart which incorporates
the KdV singularity manifold equation and the
KdV "zero soliton" equation is then
constructed. It reveals the close analogy
between the Harry Dym and Kawamoto equations
which turn out to have isomorphic symmetry
groups of symmetries. Finally, a new
integrability test is proposed which has been
termed the Expansion Test. It represents an
extension of the Painleve test. The close
connection between the singularity manifold
equation and the interacting soliton structure
follows naturally from the Expansion Test.
(Abstract shortened with permission of
author.)