Implicit similarities and inverse isovector methods
The group theoretic method, indicated by notions such as contact ideal, isovector field, similarity solution [see \textit{D. G. B. Edelen}, Isovector methods for Equations of Balance. With programs for computer assistance in operator calculations and an exposition of practical topics of the exterior calculus (1980; Zbl 0452.58001)], is applied to a system of second order p.d.e.'s in two independent and two dependent variables, which may be described by the balance 2-forms \[ (*)\quad B_{\alpha}=F^ 0_{\alpha}dx^ 1\wedge dx^ 2-d(F^ 1_{\alpha}dx^ 2-F^ 2_{\alpha}dx^ 1) \] with given functions \(F^ a_{\alpha}=F^ a_{\alpha}(x^ i,q^{\alpha},q^{\alpha}_{x_ i})\); i,\(\alpha\in \{1,2\}.\) The main part of the paper is an illustration by example of the inverse isovector problem: To find (all systems (*), whose balance ideal admits as an isovector field the prolongation of the vector field \(-x^ 1\partial /dx^ 1+(1-2q^ 1)\partial /\partial x^ 2+q^ 1\partial /\partial q^ 1+x^ 1\partial /\partial q^ 2\). This example and the corresponding reduction of (*) to a system of o.d.e.'s is carried out in detail, with special emphasis on the fact that the similarity variables occurring here are functions of the original independent and dependent variables as well.
scientific article; zbMATH DE number 3897422