1,712 research outputs found
Invariants of hyperbolic partial differential operators
We present a construction of a large class of Laplace invariants for linear hyperbolic partial differential operators of fairly general form and arbitrary order
Geometry of Darboux-Manakov-Zakharov systems and its application
The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov
systems of semilinear partial differential equations \label{GDMZabstract}
\frac{\partial^2 u}{\partial x_i\partial x_j}=f_{ij}\Big(x_k,u,\frac{\partial
u}{\partial x_l}\Big), 1\leq i<j\leq n, k,l\in\{1,...,n\} for a real-valued
function are studied with particular reference to the linear
systems in this equation class.
System (\ref{GDMZabstract}) will not generally be involutive in the sense of
Cartan: its coefficients will be constrained by complicated nonlinear
integrability conditions. We derive geometric tools for explicitly constructing
involutive systems of the form (\ref{GDMZabstract}), essentially solving the
integrability conditions. Specializing to the linear case provides us with a
novel way of viewing and solving the multi-dimensional -wave resonant
interaction system and its modified version as well as constructing new
examples of semi-Hamiltonian systems of hydrodynamic type. The general theory
is illustrated by a study of these applications
Structure of Symmetry Groups via Cartan's Method: Survey of Four Approaches
In this review article we discuss four recent methods for computing
Maurer-Cartan structure equations of symmetry groups of differential equations.
Examples include solution of the contact equivalence problem for linear
hyperbolic equations and finding a contact transformation between the
generalized Hunter-Saxton equation and the Euler-Poisson equation.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Multiple Factorizations of Bivariate Linear Partial Differential Operators
We study the case when a bivariate Linear Partial Differential Operator
(LPDO) of orders three or four has several different factorizations.
We prove that a third-order bivariate LPDO has a first-order left and right
factors such that their symbols are co-prime if and only if the operator has a
factorization into three factors, the left one of which is exactly the initial
left factor and the right one is exactly the initial right factor. We show that
the condition that the symbols of the initial left and right factors are
co-prime is essential, and that the analogous statement "as it is" is not true
for LPDOs of order four.
Then we consider completely reducible LPDOs, which are defined as an
intersection of principal ideals. Such operators may also be required to have
several different factorizations. Considering all possible cases, we ruled out
some of them from the consideration due to the first result of the paper. The
explicit formulae for the sufficient conditions for the complete reducibility
of an LPDO were found also
Invariant Discretization Schemes Using Evolution-Projection Techniques
Finite difference discretization schemes preserving a subgroup of the maximal
Lie invariance group of the one-dimensional linear heat equation are
determined. These invariant schemes are constructed using the invariantization
procedure for non-invariant schemes of the heat equation in computational
coordinates. We propose a new methodology for handling moving discretization
grids which are generally indispensable for invariant numerical schemes. The
idea is to use the invariant grid equation, which determines the locations of
the grid point at the next time level only for a single integration step and
then to project the obtained solution to the regular grid using invariant
interpolation schemes. This guarantees that the scheme is invariant and allows
one to work on the simpler stationary grids. The discretization errors of the
invariant schemes are established and their convergence rates are estimated.
Numerical tests are carried out to shed some light on the numerical properties
of invariant discretization schemes using the proposed evolution-projection
strategy
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