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Multiplication of solutions for linear overdetermined systems of partial differential equations
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2008 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 58, no 8, 1015-1029 p.Article in journal (Refereed) Published
Abstract [en]

A large family of linear, usually overdetermined, systems of partialdifferential equations that admit a multiplication of solutions, i.e, a bilinearand commutative mapping on the solution space, is studied. Thisfamily of PDE’s contains the Cauchy–Riemann equations and the cofactorpair systems, included as special cases. The multiplication provides amethod for generating, in a pure algebraic way, large classes of non-trivialsolutions that can be constructed by forming convergent power series oftrivial solutions.

Place, publisher, year, edition, pages
2008. Vol. 58, no 8, 1015-1029 p.
Keyword [en]
Overdetermined systems of PDE’s, Cauchy–Riemann equations, Power series, Superposition principle
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-14678DOI: 10.1016/j.geomphys.2008.03.008OAI: oai:DiVA.org:liu-14678DiVA: diva2:24238
Available from: 2007-10-03 Created: 2007-10-03 Last updated: 2017-12-13
In thesis
1. Systems of Linear First Order Partial Differential Equations Admitting a Bilinear Multiplication of Solutions
Open this publication in new window or tab >>Systems of Linear First Order Partial Differential Equations Admitting a Bilinear Multiplication of Solutions
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The Cauchy–Riemann equations admit a bilinear multiplication of solutions, since the product of two holomorphic functions is again holomorphic. This multiplication plays the role of a nonlinear superposition principle for solutions, allowing for construction of new solutions from already known ones, and it leads to the exceptional property of the Cauchy–Riemann equations that all solutions can locally be built from power series of a single solution z = x + iy ∈ C.

In this thesis we have found a differential algebraic characterization of linear first order systems of partial differential equations admitting a bilinear ∗-multiplication of solutions, and we have determined large new classes of systems having this property. Among them are the already known quasi-Cauchy–Riemann equations, characterizing integrable Newton equations, and the gradient equations ∇f = Mg with constant matrices M. A systematic description of linear systems of PDEs with variable coefficients have been given for systems with few independent and few dependent variables.

An important property of the ∗-multiplication is that infinite families of solutions can be constructed algebraically as power series of known solutions. For the equation ∇f = Mg it has been proved that the general solution, found by Jodeit and Olver, can be locally represented as convergent power series of a single simple solution similarly as for solutions of the Cauchy–Riemann equations.

Place, publisher, year, edition, pages
Matematiska institutionen, 2007. 20 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1135
Keyword
Cauchy–Riemann equations, holomorphic functions, algebra
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-9949 (URN)978-91-85895-78-6 (ISBN)
Public defence
2007-11-09, Glashuset, Hus B, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2007-10-03 Created: 2007-10-03 Last updated: 2009-05-04

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Jonasson, Jens

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