12 research outputs found
Group classification of variable coefficient KdV-like equations
The exhaustive group classification of the class of KdV-like equations with
time-dependent coefficients is carried out using
equivalence based approach. A simple way for the construction of exact
solutions of KdV-like equations using equivalence transformations is described.Comment: 8 pages; minor misprints are corrected. arXiv admin note: substantial
text overlap with arXiv:1104.198
Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities
A class of variable coefficient (1+1)-dimensional nonlinear
reaction-diffusion equations of the general form
is investigated. Different kinds of
equivalence groups are constructed including ones with transformations which
are nonlocal with respect to arbitrary elements. For the class under
consideration the complete group classification is performed with respect to
convenient equivalence groups (generalized extended and conditional ones) and
with respect to the set of all point transformations. Usage of different
equivalences and coefficient gauges plays the major role for simple and clear
formulation of the final results. The corresponding set of admissible
transformations is described exhaustively. Then, using the most direct method,
we classify local conservation laws. Some exact solutions are constructed by
the classical Lie method.Comment: 23 pages, minor misprints are correcte
Equivalence of conservation laws and equivalence of potential systems
We study conservation laws and potential symmetries of (systems of)
differential equations applying equivalence relations generated by point
transformations between the equations. A Fokker-Planck equation and the Burgers
equation are considered as examples. Using reducibility of them to the
one-dimensional linear heat equation, we construct complete hierarchies of
local and potential conservation laws for them and describe, in some sense, all
their potential symmetries. Known results on the subject are interpreted in the
proposed framework. This paper is an extended comment on the paper of J.-q. Mei
and H.-q. Zhang [Internat. J. Theoret. Phys., 2006, in press].Comment: 10 page
A precise definition of reduction of partial differential equations
We give a comprehensive analysis of interrelations between the basic concepts
of the modern theory of symmetry (classical and non-classical) reductions of
partial differential equations. Using the introduced definition of reduction of
differential equations we establish equivalence of the non-classical
(conditional symmetry) and direct (Ansatz) approaches to reduction of partial
differential equations. As an illustration we give an example of non-classical
reduction of the nonlinear wave equation in (1+3) dimensions. The conditional
symmetry approach when applied to the equation in question yields a number of
non-Lie reductions which are far-reaching generalization of the well-known
symmetry reductions of the nonlinear wave equations.Comment: LaTeX, 21 page
Exact Solutions of a Remarkable Fin Equation
A model "remarkable" fin equation is singled out from a class of nonlinear
(1+1)-dimensional fin equations. For this equation a number of exact solutions
are constructed by means of using both classical Lie algorithm and different
modern techniques (functional separation of variables, generalized conditional
symmetries, hidden symmetries etc).Comment: 6 page
Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source
A new approach to group classification problems and more general
investigations on transformational properties of classes of differential
equations is proposed. It is based on mappings between classes of differential
equations, generated by families of point transformations. A class of variable
coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the
general form () is studied from the
symmetry point of view in the framework of the approach proposed. The singular
subclass of the equations with is singled out. The group classifications
of the entire class, the singular subclass and their images are performed with
respect to both the corresponding (generalized extended) equivalence groups and
all point transformations. The set of admissible transformations of the imaged
class is exhaustively described in the general case . The procedure of
classification of nonclassical symmetries, which involves mappings between
classes of differential equations, is discussed. Wide families of new exact
solutions are also constructed for equations from the classes under
consideration by the classical method of Lie reductions and by generation of
new solutions from known ones for other equations with point transformations of
different kinds (such as additional equivalence transformations and mappings
between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica
Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited
We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w
w_x + (2-4\alpha)z z_x, z_t &=& (1-2\alpha)z_{xx} - 4\alpha z w_x +
(4-8\alpha)w z_x - (4+8\alpha)w^2 z + (-2+4\alpha)z^3, (*) for which no
recursion operator or master symmetry was known so far, and prove that the
system (*) admits infinitely many local generalized symmetries that are
constructed using a nonlocal {\em two-term} recursion relation rather than from
a recursion operator.Comment: 10 pages, LaTeX; minor changes in terminology; some references and
definitions adde
A Precise Definition Of Reduction Of Partial Differential Equations
We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in 1+3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalization of the well-known symmetry reductions of the nonlinear wave equations