1,473 research outputs found
Connections Between Symmetries and Conservation Laws
This paper presents recent work on connections between symmetries and
conservation laws. After reviewing Noether's theorem and its limitations, we
present the Direct Construction Method to show how to find directly the
conservation laws for any given system of differential equations. This method
yields the multipliers for conservation laws as well as an integral formula for
corresponding conserved densities. The action of a symmetry (discrete or
continuous) on a conservation law yields conservation laws. Conservation laws
yield non-locally related systems that, in turn, can yield nonlocal symmetries
and in addition be useful for the application of other mathematical methods.
From its admitted symmetries or multipliers for conservation laws, one can
determine whether or not a given system of differential equations can be
linearized by an invertible transformation.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Continuous and discrete transformations of a one-dimensional porous medium equation
We consider the one-dimensional porous medium equation . We derive point transformations of a general
class that map this equation into itself or into equations of a similar class.
In some cases this porous medium equation is connected with well known
equations. With the introduction of a new dependent variable this partial
differential equation can be equivalently written as a system of two equations.
Point transformations are also sought for this auxiliary system. It turns out
that in addition to the continuous point transformations that may be derived by
Lie's method, a number of discrete transformations are obtained. In some cases
the point transformations which are presented here for the single equation and
for the auxiliary system form cyclic groups of finite order
Lie symmetries and conservation laws of the Hirota-Ramani equation
In this paper, Lie symmetry method is performed for the Hirota-Ramani (H-R)
equation. We will find The symmetry group and optimal systems of Lie
subalgebras. Furthermore, preliminary classification of its group invariant
solutions, symmetry reduction and nonclassical symmeries are investigated.
Finally the conservation laws of the H-R equation are presented.Comment: 16 page
Classical and nonclassical symmetries of a generalized Boussinesq equation
We apply the Lie-group formalism and the nonclassical method due to Bluman
and Cole to deduce symmetries of the generalized Boussinesq equation, which has
the classical Boussinesq equation as an special case. We study the class of
functions for which this equation admit either the classical or the
nonclassical method. The reductions obtained are derived. Some new exact
solutions can be derived
Conservation laws for nonlinear telegraph equations
AbstractA complete conservation law classification is given for nonlinear telegraph (NLT) systems with respect to multipliers that are functions of independent and dependent variables. It turns out that a very large class of NLT systems admits four nontrivial local conservation laws. The results of this work are summarized in tables which display all multipliers, fluxes and densities for the corresponding conservation laws. A physical example is considered for possible applications
Lie symmetries and exact solutions of variable coefficient mKdV equations: an equivalence based approach
Group classification of classes of mKdV-like equations with time-dependent
coefficients is carried out. The usage of equivalence transformations appears a
crucial point for the exhaustive solution of the problem. We prove that all the
classes under consideration are normalized. This allows us to formulate the
classification results in three ways: up to two kinds of equivalence (which are
generated by transformations from the corresponding equivalence groups and all
admissible point transformations) and using no equivalence. A simple way of the
construction of exact solutions of mKdV-like equations using equivalence
transformations is described.Comment: 10 pages, minor change
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