Multiplication for solutions of the equation grad f = M grad g
2009 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, Vol. 59, no 10, 1412-1430 p.Article in journal (Refereed) Published
Linear first-order systems of partial differential equations (PDEs) of the form ∇ f = M ∇ g, where M is a constant matrix, are studied on vector spaces over the fields of real and complex numbers. The Cauchy-Riemann equations belong to this class. We introduce on the solution space a bilinear *-multiplication, playing the role of a nonlinear superposition principle, that allows for algebraic construction of new solutions from known solutions. The gradient equation ∇ f = M ∇ g is a simple special case of a large class of systems of PDEs, admitting a *-multiplication of solutions. We prove that any gradient equation has the exceptional property that the general analytic solution can be expressed as *-power series of certain simple solutions.
Place, publisher, year, edition, pages
2009. Vol. 59, no 10, 1412-1430 p.
Cauchy-Riemann equations; Multiplication of solutions; Overdetermined systems of PDEs; Power series
IdentifiersURN: urn:nbn:se:liu:diva-21201DOI: 10.1016/j.geomphys.2009.07.006OAI: oai:DiVA.org:liu-21201DiVA: diva2:240897