30,236 research outputs found
Differential constraints and exact solutions of nonlinear diffusion equations
The differential constraints are applied to obtain explicit solutions of
nonlinear diffusion equations. Certain linear determining equations with
parameters are used to find such differential constraints. They generalize the
determining equations used in the search for classical Lie symmetries
On the Integrability, B\"Acklund Transformation and Symmetry Aspects of a Generalized Fisher Type Nonlinear Reaction-Diffusion Equation
The dynamics of nonlinear reaction-diffusion systems is dominated by the
onset of patterns and Fisher equation is considered to be a prototype of such
diffusive equations. Here we investigate the integrability properties of a
generalized Fisher equation in both (1+1) and (2+1) dimensions. A Painlev\'e
singularity structure analysis singles out a special case () as
integrable. More interestingly, a B\"acklund transformation is shown to give
rise to a linearizing transformation for the integrable case. A Lie symmetry
analysis again separates out the same case as the integrable one and
hence we report several physically interesting solutions via similarity
reductions. Thus we give a group theoretical interpretation for the system
under study. Explicit and numerical solutions for specific cases of
nonintegrable systems are also given. In particular, the system is found to
exhibit different types of travelling wave solutions and patterns, static
structures and localized structures. Besides the Lie symmetry analysis,
nonclassical and generalized conditional symmetry analysis are also carried
out.Comment: 30 pages, 10 figures, to appear in Int. J. Bifur. Chaos (2004
Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification
We discuss the classical statement of group classification problem and some
its extensions in the general case. After that, we carry out the complete
extended group classification for a class of (1+1)-dimensional nonlinear
diffusion--convection equations with coefficients depending on the space
variable. At first, we construct the usual equivalence group and the extended
one including transformations which are nonlocal with respect to arbitrary
elements. The extended equivalence group has interesting structure since it
contains a non-trivial subgroup of non-local gauge equivalence transformations.
The complete group classification of the class under consideration is carried
out with respect to the extended equivalence group and with respect to the set
of all point transformations. Usage of extended equivalence and correct choice
of gauges of arbitrary elements play the major role for simple and clear
formulation of the final results. The set of admissible transformations of this
class is preliminary investigated.Comment: 25 page
A variational principle for computing slow invariant manifolds in dissipative dynamical systems
A key issue in dimension reduction of dissipative dynamical systems with
spectral gaps is the identification of slow invariant manifolds. We present
theoretical and numerical results for a variational approach to the problem of
computing such manifolds for kinetic models using trajectory optimization. The
corresponding objective functional reflects a variational principle that
characterizes trajectories on, respectively near, slow invariant manifolds. For
a two-dimensional linear system and a common nonlinear test problem we show
analytically that the variational approach asymptotically identifies the exact
slow invariant manifold in the limit of both an infinite time horizon of the
variational problem with fixed spectral gap and infinite spectral gap with a
fixed finite time horizon. Numerical results for the linear and nonlinear model
problems as well as a more realistic higher-dimensional chemical reaction
mechanism are presented.Comment: 16 pages, 5 figure
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