67,727 research outputs found
Potential Conservation Laws
We prove that potential conservation laws have characteristics depending only
on local variables if and only if they are induced by local conservation laws.
Therefore, characteristics of pure potential conservation laws have to
essentially depend on potential variables. This statement provides a
significant generalization of results of the recent paper by Bluman, Cheviakov
and Ivanova [J. Math. Phys., 2006, V.47, 113505]. Moreover, we present
extensions to gauged potential systems, Abelian and general coverings and
general foliated systems of differential equations. An example illustrating
possible applications of proved statements is considered. A special version of
the Hadamard lemma for fiber bundles and the notions of weighted jet spaces are
proposed as new tools for the investigation of potential conservation laws.Comment: 36 pages, extended versio
Conditional symmetries and Riemann invariants for inhomogeneous hydrodynamic-type systems
A new approach to the solution of quasilinear nonelliptic first-order systems
of inhomogeneous PDEs in many dimensions is presented. It is based on a version
of the conditional symmetry and Riemann invariant methods. We discuss in detail
the necessary and sufficient conditions for the existence of rank-2 and rank-3
solutions expressible in terms of Riemann invariants. We perform the analysis
using the Cayley-Hamilton theorem for a certain algebraic system associated
with the initial system. The problem of finding such solutions has been reduced
to expanding a set of trace conditions on wave vectors and their profiles which
are expressible in terms of Riemann invariants. A couple of theorems useful for
the construction of such solutions are given. These theoretical considerations
are illustrated by the example of inhomogeneous equations of fluid dynamics
which describe motion of an ideal fluid subjected to gravitational and Coriolis
forces. Several new rank-2 solutions are obtained. Some physical interpretation
of these results is given.Comment: 19 page
Representing a nonlinear state space system as a set of higher-order differential equations in the inputs and ouputs
An algorithm is presented for transforming a nonlinear state space system into a threefold set of equations; the first subset describing the dynamics of the unobservable part of the system, the second subset representing the remaining state variables as functions of inputs and outputs and their derivatives, and the last subset defining the external behaviour of the system
On the Index and the Order of Quasi-regular Implicit Systems of Differential Equations
This paper is mainly devoted to the study of the differentiation index and
the order for quasi-regular implicit ordinary differential algebraic equation
(DAE) systems. We give an algebraic definition of the differentiation index and
prove a Jacobi-type upper bound for the sum of the order and the
differentiation index. Our techniques also enable us to obtain an alternative
proof of a combinatorial bound proposed by Jacobi for the order.
As a consequence of our approach we deduce an upper bound for the
Hilbert-Kolchin regularity and an effective ideal membership test for
quasi-regular implicit systems. Finally, we prove a theorem of existence and
uniqueness of solutions for implicit differential systems
Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations in Multi-dimensions
A direct method for the computation of polynomial conservation laws of
polynomial systems of nonlinear partial differential equations (PDEs) in
multi-dimensions is presented. The method avoids advanced
differential-geometric tools. Instead, it is solely based on calculus,
variational calculus, and linear algebra.
Densities are constructed as linear combinations of scaling homogeneous terms
with undetermined coefficients. The variational derivative (Euler operator) is
used to compute the undetermined coefficients. The homotopy operator is used to
compute the fluxes.
The method is illustrated with nonlinear PDEs describing wave phenomena in
fluid dynamics, plasma physics, and quantum physics. For PDEs with parameters,
the method determines the conditions on the parameters so that a sequence of
conserved densities might exist. The existence of a large number of
conservation laws is a predictor for complete integrability. The method is
algorithmic, applicable to a variety of PDEs, and can be implemented in
computer algebra systems such as Mathematica, Maple, and REDUCE.Comment: To appear in: Thematic Issue on ``Mathematical Methods and Symbolic
Calculation in Chemistry and Chemical Biology'' of the International Journal
of Quantum Chemistry. Eds.: Michael Barnett and Frank Harris (2006
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