Lecture Notes on Geometrical Aspects of Partial Differential Equations

A remark on nonlinear accessibility conditions and infinite prolongations

“…We remark here that the strong and weak accessibility Lie rank conditions possess an easy and nice interpretation within the framework of the differential geometry of jets and prolongations of infinite order, which is becoming a mainstay in some parts of mathematics and physics (see, e.g., [3,4,32,38,41,49,66,67,69] and the references therein). Motivated by the study of differential flatness [25], i.e., by dynamic feedback linearizabilty, this infinite-dimensional geometry is now being developed in nonlinear control (cf.…”

“…, n; ν i ≥ 0}, where the total derivative d dt with respect to the time t is given by the following infinite vector field, called a Cartan field (cf. [41,69]),…”

“…[26]). We thus introduce the notion of diffiety, i.e., of an abstract infinite-dimensional manifold equipped with a Cartan distribution (see [41,65,66,67,69]), which is justified by an example. We show that an orbitally flat system, which is Lie-Bäcklund isomorphic to a trivial diffiety, does not possess any non-trivial local first integral.…”

“…, u (ν) ) on D it yields its total derivative with respect to t: d dt is called a Cartan field (cf. [41,69]). The tangent distribution of D is spanned (iv) The only local first integrals on D are the trivial ones.…”

Об Обратимых Преобразованиях Беклунда Для Автономных Эволюционных Уравнений

“…При построении интегрирующих множителей и законов сохранения мы следуем работам [4]- [8], применяя обозначения из работ [9], [10] (относительно симметрийных интегрируемых уравнений эволюции см. статьи [11], [12] и ссылки в этих работах).…”