|
Content
Research in progress and
future directions Starting
from investigating multiple solutions for the Henon Equation
via a Newton Homotopy Continuation Method, we have now shifted our
gears to the
computational theory and numerical methods on
multiple (unstable) solutions to nonlinear elliptic problems including cooperative and noncooperative elliptic
systems, Hamiltonian
elliptic systems(e.g., the Lane-Emden system, semilinear
biharmonic problems). Instability analysis of the solutions is also among the scope of our study. Since the Morse index (refer to my
favourite torus or the Morse theory)
plays a significant role in the Critical Point Theory, estimates of
the Morse index or relative Morse index (arising in noncooperative and
Hamiltonian cases) need to be established. While we will continue
our study on
noncooperative and Hamiltonian cases, our long-term goal is to extend the methods we
have developed to more challenging nonlinear problems such as nonvariational systems, nonlinear
wave equations
(or even systems of coupled nonlinear wave equations), Hamiltonian
PDE systems (often referred to as infinite dimensional Hamiltonian
systems) and Hamiltonian
ODE
systems (often referred to as finite dimensional Hamiltonian
systems or Hamilton
equations) which are not necessarily separable and/or autonomous
(time-independent).
We note that there are a number of symplectic
or splitting methods as
well as relavant solvers available now, most of which are mainly
devoted
to solving separable and/or autonomous Hamiltonian systems (see,
e.g., "Symplectic
Methods for Separable Hamiltonian Systems" by Mark Sofroniou and Giulia
Spaletta).
For general Hamiltonian systems (non-separable and/or non-autonomous),
many challenges arise and need to be overcome,
such as how
to eliminate or reduce the effect of round-off errors,
how to preserve the symplectic structure while conserving the energy
(unfortunately, it has been shown that this is in general impossible,
see, e.g., Symplectic
Methods for Hamiltonian Systems). By incorporating minimax and variational methods
and/or
techniques, we shall develop some
reliable methods to solve them.
A motivation on finding
unstable solutions When
problems to be solved are
variational, they can be viewed
as the Euler-Lagrange
equations of certain functionals whose critical
points correspond to
their weak solutions under some standard hypothesis. Critical points
that are local extrema (i.e., maxima or minima)
are the main subject of traditional critical point theory and
well-studied in calculus
of variations since 1743 (initiated by Euler).
Critical points correponding
neither
to local maxima nor to local
minima are the
so-called saddle
points
which appear as unstable equilibria or transient excited states in
physical systems, and are usually thought
to be too hard to reach or obtain in applications. With new atomic,
optical and/or synchrotronic technologies, however, scientists now are
able to successfully trap or secure them and start to search for their
very promising applications. These new technologies and other advanced
studies in related areas are changing the traditional view on unstable
solutions and starting to draw more and more attention. As a result,
there is a
growing need to develop more efficient and reliable numerical methods
for finding those unstable
solutions (i.e.,
excited states) in order to get a better
understanding of their intrinsic behaviors
and dynamics, since analytic
solutions are in general too difficult to obtain.
A
short history on critical point theory Followed
by the work of Poincare
who made a valuable contribution to
the calculus of variations in 1905, Birkhoff succeeded to obtain a
minimax principle where a saddle point
is characterized as a minimax in 1917. This serves as the beginning of
modern critical point theory which is
devoted to studies of saddle points. A theory of minimax was elaborated
in the
late 1920s and early 1930s independently by Morse
and by Ljusternik and Schnirelman. Their results contain the basic
ingredients of modern minimax theorems
and form a substantial part of modern
critical point theory. The well-known Mountain
Pass Theorem, proved in 1973 by Ambrosetti and Rabinowitz, set undoubtedly a
milestone in critical
point
theory. It influenced its "successors" so much that it can be
considered as the beginning of a postmodern era in this theory. The
theorem was demonstrated to be so useful and was repeatedly used as
a model or framework for a number of other critical point theorems
(such as Saddle Point Theorem, Linking Theorem, etc)
followed.
|