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Systems of Linear First Order Partial Differential Equations Admitting a Bilinear Multiplication of Solutions
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
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 [en]
Cauchy–Riemann equations, holomorphic functions, algebra
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-9949ISBN: 978-91-85895-78-6 (print)OAI: oai:DiVA.org:liu-9949DiVA: diva2:24241
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
List of papers
1. The equation X ∇det X = det X ∇ trX multiplication of cofactor pair systems, and the Levi-Civita equivalence problem
Open this publication in new window or tab >>The equation X ∇det X = det X ∇ trX multiplication of cofactor pair systems, and the Levi-Civita equivalence problem
2006 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, Vol. 57, no 1, 251-267 p.Article in journal (Refereed) Published
Abstract [en]

Cofactor pair systems generalize the separable potential Hamiltonian systems. They admit n quadratic integrals of motion, they have a bi-Hamilton formulation, they are completely integrable and they are equivalent to separable Lagrangian systems. Cofactor pair systems can be constructed through a peculiar multiplicative structure of the so-called quasi-Cauchy–Riemann equations , where J and are special conformal Killing tensors.

In this work we have isolated the properties that are responsible for the multiplication, allowing us to give an elegant characterization of systems that admit multiplication. In this characterization the equation plays a central role.

We describe how multiplication of quasi-Cauchy–Riemann equations can be considered as a special case of a more general kind of multiplication, defined on the solution space of certain systems of partial differential equations. We investigate algebraic properties of this multiplication and provide several methods for constructing new systems with multiplicative structure. We also discuss the role of the multiplication in the theory of equivalent dynamical systems on Riemannian manifolds, developed by Levi-Civita.

Keyword
Killing tensor; Nijenhuis torsion; Cauchy–Riemann equations; Separation of variables
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-14677 (URN)10.1016/j.geomphys.2006.03.001 (DOI)
Available from: 2007-10-03 Created: 2007-10-03
2. Multiplication of solutions for linear overdetermined systems of partial differential equations
Open this publication in new window or tab >>Multiplication of solutions for linear overdetermined systems of partial differential equations
2008 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, 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.

Keyword
Overdetermined systems of PDE’s, Cauchy–Riemann equations, Power series, Superposition principle
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-14678 (URN)10.1016/j.geomphys.2008.03.008 (DOI)
Available from: 2007-10-03 Created: 2007-10-03 Last updated: 2009-05-04
3. Multiplication for solutions of the equation grad f = M grad g
Open this publication in new window or tab >>Multiplication for solutions of the equation grad f = M grad g
Manuscript (Other academic)
Identifiers
urn:nbn:se:liu:diva-14679 (URN)
Available from: 2007-10-03 Created: 2007-10-03 Last updated: 2010-01-13
4. An explicit formula for the polynomial remainder using the companion matrix of the divisor
Open this publication in new window or tab >>An explicit formula for the polynomial remainder using the companion matrix of the divisor
Manuscript (Other academic)
Identifiers
urn:nbn:se:liu:diva-14680 (URN)
Available from: 2007-10-03 Created: 2007-10-03 Last updated: 2010-01-13

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Jonasson, Jens

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