25 research outputs found
New conditional symmetries and exact solutions of nonlinear reaction-diffusion-convection equations. II
In the first part of this paper math-ph/0612078, a complete description of
Q-conditional symmetries for two classes of reaction-diffusion-convection
equations with power diffusivities is derived. It was shown that all the known
results for reaction-diffusion equations with power diffusivities follow as
particular cases from those obtained in math-ph/0612078 but not vise versa. In
the second part the symmetries obtained in are successfully applied for
constructing exact solutions of the relevant equations. In the particular case,
new exact solutions of nonlinear reaction-diffusion-convection (RDC) equations
arising in application and their natural generalizations are found
Conservation laws for self-adjoint first order evolution equations
In this work we consider the problem on group classification and conservation
laws of the general first order evolution equations. We obtain the subclasses
of these general equations which are quasi-self-adjoint and self-adjoint. By
using the recent Ibragimov's Theorem on conservation laws, we establish the
conservation laws of the equations admiting self-adjoint equations. We
illustrate our results applying them to the inviscid Burgers' equation. In
particular an infinite number of new symmetries of these equations are found
and their corresponding conservation laws are established.Comment: This manuscript has been accepted for publication in Journal of
Nonlinear Mathematical Physic
New variable separation approach: application to nonlinear diffusion equations
The concept of the derivative-dependent functional separable solution, as a
generalization to the functional separable solution, is proposed. As an
application, it is used to discuss the generalized nonlinear diffusion
equations based on the generalized conditional symmetry approach. As a
consequence, a complete list of canonical forms for such equations which admit
the derivative-dependent functional separable solutions is obtained and some
exact solutions to the resulting equations are described.Comment: 19 pages, 2 fig
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
Symmetries and currents of the ideal and unitary Fermi gases
The maximal algebra of symmetries of the free single-particle Schroedinger
equation is determined and its relevance for the holographic duality in
non-relativistic Fermi systems is investigated. This algebra of symmetries is
an infinite dimensional extension of the Schroedinger algebra, it is isomorphic
to the Weyl algebra of quantum observables, and it may be interpreted as a
non-relativistic higher-spin algebra. The associated infinite collection of
Noether currents bilinear in the fermions are derived from their relativistic
counterparts via a light-like dimensional reduction. The minimal coupling of
these currents to background sources is rewritten in a compact way by making
use of Weyl quantisation. Pushing forward the similarities with the holographic
correspondence between the minimal higher-spin gravity and the critical O(N)
model, a putative bulk dual of the unitary and the ideal Fermi gases is
discussed.Comment: 67 pages, 2 figures; references added, minor improvements in the
presentation, version accepted for publication in JHE
Exact nonclassical symmetry solutions of Lotka-Volterra-type population systems
New classes of conditionally integrable systems of nonlinear reaction-diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator-prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka-Volterra system, but they have additional features