Direct construction method for conservation laws of partial differential equations Part II: General treatment

Abstract

This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.

Об Обратимых Преобразованиях Беклунда Для Автономных Эволюционных Уравнений

“…Следуя работе [10], имеем Предложение 1. Пусть Λ -присоединенная симметрия уравнения (2.1), ко-торая проявляет себя в виде интегрирующего множителя для (2.1), т.е.…”

“…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).…”

Symbolic computation of conservation laws of nonlinear partial differential equations in multi‐dimensions

“…If we select W (u) = W (v) = W (w) = a = 1 4 , then the first three densities have rank 1 4 and ρ (4) , ρ (5) , and ρ (6) have ranks 1 2 , 1, 5 2 , respectively. The physics of (6) demand that the magnitude S = ||S|| of the spin field is constant in time.…”

“…Nonetheless, we draw the reader's attention to DE_APPLS, a package for constructing conservation laws from symmetries available within Vessiot [7], a general purpose suite of Maple packages for computations on jet spaces. Other methods circumvent the existence of a variational principle [4,5,29,30] but still rely on the symbolic solution of a determining system of PDEs. Despite their power, only a few of the above methods have been fully implemented in CAS (see [16,19,29]).…”

“…Indeed, the homotopy operator reduces that problem to a standard one-dimensional integration with respect to a single auxiliary parameter. One of the first 1 uses of the homotopy operator in the context of conservation laws can be found in [5]. The homotopy operator is a universal, yet little known, tool that can be applied to many problems in which integration by parts in multi-variables is needed.…”