The governing equations of dynamic meteorology are generally nonlinear partial differential equations. They describe the evolution of the atmospheric variables, including the velocity, vorticity and divergence fields. Due to the high complexity of interaction of these fields the overall dominating way to obtain solutions of these equations is numerical integration. For conceptual understanding, however, as well as for benchmark tests of numerical weather and climate prediction models, exact solutions of the underlying equations are highly desirable. In this talk, we aim to give examples of two methods to attack this problem, exemplified with the inviscid barotropic vorticity equation (IBVE). The IBVE qualitatively describes the behaviour of large-scale, two-dimensional fluid dynamics of mid-latitudes of the rotating earth. We will firstly present group-invariant solutions of the IBVE, which include often observed planetary waves. Secondly, we will use induced symmetries of the Fourier transformed version of the IBVE to obtain the minimal system of coupled ordinary differential equations, derived by Edward Lorenz as a simple model of interaction of a zonal flow with disturbances of this flow. This method may be an example of a general way how to obtain consistent finite-mode approximations of partial differential equations.

Sergey Golovin
Hierarchy of partially invariant solutions to differential equations

Abstract

It is noticed, that the partially invariant solution (PIS) of a system of differential equations in many cases can be represented as an invariant reduction of some PIS of the higher rank. This introduces a hierarchic structure in the set of all PISs of a given system of differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. The hierarchy simplifies the process of enumeration and symbolic analysis of partially invariant submodels to the given system of differential equations. This approach is demonstrated on examples of the hierarchies of regular PISs to shallow water equations and ideal MHD equations.

Kesh Govinder (joint work with Barbara Abraham-Shrauner)
Hidden symmetries of PDEs

Abstract

Type II hidden symmetries of partial differential equations (PDEs) are point symmetries that arise unexpectedly when the number of variables of PDEs are decreased. Here we show how to determine the origin of these hidden symmetries as point symmetries of a master PDE. In addition, we show that they arise from a number of different sources which include potential and contact symmetries.

Vladimir Kornyak
Gauge invariance in discrete dynamical systems

Abstract (.pdf file)

The fundamental laws of physics are described from the modern point of view by gauge theories with continuous gauge groups. With some modification the gauge principle can (and should) be applied to discrete dynamical systems with nontrivial symmetries also. We consider specifics of introduction of gauge invariance into W-equivariant discrete dynamical systems with finite sets of states , where W is the group of symmetries of . Typical examples of such systems are cellular automata on finite lattices… (see the pdf file for the complete abstract)

Michal Marvan

On zero curvature representations, the spectral parameter problem, and recursion operators

DVI File

Abstract

The talk will discuss algorithmic aspects of classification of PDE possessing a zero curvature representation with values in a given Lie algebra. In full generality, the problem reduces to a quasilinear system of equations in total derivatives.

Zero curvature representations depending on a nonremovable parameter (spectral parameter) are among the most important attributes of two-dimensional integrable systems. In many cases a given system comes with a parameterless zero curvature representation. The problem whether a parameter can be incorporated will be called the spectral parameter problem.

We shall show how to reduce the spectral parameter problem to a system of linear equations in total derivatives. In case when the solution lies in the same Lie algebra, we have a computable cohomological obstruction to existence of a nonremovable parameter. On the other hand, even a parameterless zero curvature representation usually allows us to find the recursion operator, hence to insert the parameter.

Sergey V. Meleshko

Applications of Reduce in equivalence problems of differential equations

Abstract (.pdf file)

Compatibility analysis is one of the main tools of studying differential equations. One of the features of the compatibility analysis is the extensive analytical manipulations involved in the calculations. These manipulations consist of sequentially executing such operations as prolongations of a system, substitution of complicated expressions, and analysis of independency. The cumbersome part of these calculations (or certainly part of it) can be entrusted to a computer. A large body of literature exists on the topic of application of computer algebra systems. The presentation is devoted to review my own experience of applications of computer symbolic manipulations in problems related with the compatibility analysis. In the presentation these applications are roughly separated in the three parts: (a) total compatibility analysis; (b) group analysis; (c) equivalence problems of differential equations. The first two parts are shortly reviewed. New results obtained in the third part are mostly considered in the presentation. The new results include the following: (1) equivalence of second-order ordinary differential equations to canonical set of equations (equations admitting a Lie group or Painlevé equations); (2) linearization problem of third and fourth order ordinary differential equations; (3) equivalence of parabolic partial differential equations to a canonical set of equations. These problems are characterized by similar approach of solving. Their solutions are obtained by using Reduce symbolic system of manipulations.

Johannes Middeke

A polynomial time algorithm for computing the Jacobson form for matrices of differential operators

Abstract

We consider the ring R = K[\partial; \id, \theta] of differential operators over a differential field (K,\theta). It is well-known that, if K has characteristic zero, matrices over R can be transformed into Jacobson form diag(1,...1,f,0,...,0). Yet the classical algorithms seem to be of exponential time complexity. In this talk we present a new polynomial-time algorithm using modular methods. Furthermore, using this algorithm we will achieve the form diag(1,...,1,f,0,...,0) also for fields of non-zero characteristic.

Anatoly Nikitin
Symmetries in surface diffusion processes and modelling functions for coverage profiles

Abstract

An approach to theory of diffusion processes is proposed, which is based on application of both the symmetry analysis and method of modelling functions. An algorithm for construction of the modelling functions for coverage profiles is suggested, which is based on the error functions expansion (ERFE) of concentration curves. This approach is used to analysis experimental results obtained for surface diffusion of Li absorbed by Mo(112), but can be applied to many other diffusion systems.

Juha Pohjanpelto

Pseudogroups, Moving Frames, and Exterior Differential Systems

PDF File

Abstract

Continuous pseudogroups appear as the infinite dimensional counterparts of local Lie groups of transformations in various physical and geometrical contexts, including gauge theories, Hamiltonian mechanics and symplectic and Poisson geometry, conformal field theory, symmetry groups partial differential equations, such as the Navier-Stokes and Kadomtsev-Petviashvili equations of fluid mechanics and plasma physics, image recognition, and geometric numerical integration. In this talk I will report on my recent joint work with Peter Olver on extending the classical moving frames method to infinite dimensional pseudogroups. As in the finite dimensional case, moving frames can be employed to produce complete sets of differential invariants for a pseudogroup action and to effectively analyze the algebraic structure of such invariants. Moreover, I will discuss a novel reduction method based on the moving frames algorithm for exterior differential systems invariant under the action of a continuous pseudogroup and describe its applications to constructing analytic solutions to systems of partial differential equations.

Roman Popovych

Singular reduction operators

PDF File

Abstract

The singular reduction operators, i.e., the singular operators of nonclassical (conditional) symmetry, of partial differential equations and all the possible reductions of these equations to first-order ordinary differential ones are exhaustively described. The separation of singular families of reduction operators is shown to be an important step of algorithmic construction of nonclassical symmetries with symbolic calculation systems.

Ekaterina Shemyakova

Moving Frames for Laplace Invariants

Abstract

The development of symbolic methods for the factorization and integration of linear PDEs, many of the methods being generalizations of the Laplace transformations method, requires the finding of complete generating sets of invariants for the corresponding linear operators and their systems with respect to the gauge transformations. Within the theory of Laplace-like methods, there is no uniform approach to this problem, though some individual invariants for hyperbolic bivariate operators, and complete generating sets of invariants for second- and third-order hyperbolic bivariate ones have been obtained. We show a systematic and much more efficient approach to the same problem by application of moving-frame methods. We give explicit formulae for complete generating sets of invariants for second- and third-order bivariate linear operators, hyperbolic and non-hyperbolic, and also demonstrate the approach for pairs of operators appearing in Darboux transformations. Some applications to factorizations of Linear Partial Differential Operators will be discussed also.

Thomas Wolf (joint work with Sergey Tsarev)

Hyperdeterminants as integrable 3D difference equations

PDF File

Abstract

In this talk, co-authored with Sergey Tsarev, we give the basic definitions and known theoretical results about hyperdeterminants, briefly sketch their modern applications (quantum computing, biomathematics, numerics, stochastic calculus etc.) and give a proof of integrability (understood as 4d-consistency) of a difference equation defined by the 2x2x2 hyperdeterminant. Independently integrability of the 2x2x2 hyperdeterminant was stated by R.Kashaev(1996) and W.Schief (2003).

A natural conjecture is that all 2^n hyperdeterminants define discrete integrable equations.

We prove that this conjecture already fails in the case of the 2x2x2x2 hyperdeterminant, computed recently by B.Sturmfels et. al (www.arxiv.org, math.CO/0602149 v2 3 Oct 2006).

Nicoleta Bila

A special class of symmetry reductions for PDEs involving arbitrary functions – PDF

PDF File

Abstract

Special symmetry reductions for PDEs involving arbitrary functions depending only on the independent variables are discussed. These group transformations can also be associated with parameter identification problems described by PDEs. In fact, these transformations called the extended classical symmetries of the model are the classical Lie symmetries related to the equation in which both the parameter and the data are regarded as dependent variables. The relationship between these transformations and the weak equivalence transformations (due to Torrisi and Traciná), the generalized equivalence transformations (introduced by Meleshko), and the equivalence transformations in the sense of Ovsiannikov is studied. For this type of equations, any generalized equivalence transformation is an extended classical symmetry and, excepting a particular case, the extended classical symmetries differ from the weak equivalence transformations. The determining equations of the generalized equivalence transformations can be obtained from the determining equations of the extended classical symmetries. Therefore, for these equations, any symbolic manipulation program designed to find the classical Lie symmetries can also be used to determine the generalized equivalence transformations.

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Last updated: September 18, 2008