Direct construction method for conservation laws of partial differential equations. Part II: General treatment
Stephen C. Anco,George W. Bluman +1 more
TLDR
In this article, the authors give a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]), and apply it to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form.Abstract:
This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [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.read more
Citations
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Journal ArticleDOI
Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians
Abdul H. Kara,Fazal M. Mahomed +1 more
TL;DR: In this article, the authors show how to construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what they term partial Lagrangians.
Journal ArticleDOI
GeM software package for computation of symmetries and conservation laws of differential equations
TL;DR: The package contains a collection of powerful easy to use routines for mathematicians and applied researchers for automated symmetry and conservation law analysis of systems of partial and ordinary difierential equations (DE).
Journal ArticleDOI
Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics
TL;DR: The conservation laws for the non-linear diffusion equation for the spreading of an axisymmetric thin liquid drop, the system of two partial differential equations governing flow in a laminar two-dimensional jet and the system
Journal ArticleDOI
Localized Hexagon Patterns of the Planar Swift–Hohenberg Equation
TL;DR: It is found that stationary spatially localized hexagon patterns of the two-dimensional (2D) Swift–Hohenberg equation exhibit snaking: for each parameter value in the snaking region, an infinite number of patterns exist that are connected in parameter space and whose width increases without bound.
Book ChapterDOI
Generalization of Noether’s Theorem in Modern Form to Non-variational Partial Differential Equations
TL;DR: In this paper, a general method using multipliers for finding the conserved integrals admitted by any given partial differential equation (PDE) or system of partial differential equations is reviewed and further developed in several ways.
References
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Book
Applications of Lie Groups to Differential Equations
TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
Journal ArticleDOI
Symplectic structures, their Bäcklund transformations and hereditary symmetries
TL;DR: In this paper, it was shown that compatible symplectic structures lead in a natural way to hereditary symmetries, and that a hereditary symmetry is an operator-valued function which immediately yields a hierarchy of evolution equations, each having infinitely many commuting symmetry all generated by this hereditary symmetry.
Journal ArticleDOI
Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications
Stephen C. Anco,George W. Bluman +1 more
TL;DR: In this paper, an algorithm for finding local conservation laws for partial differential equations with any number of independent and dependent variables is presented, which does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations.
Journal ArticleDOI
k‐symplectic structures
TL;DR: In this article, the basic properties of k-symplectic Lie algebras are introduced and developed in analogy with the well-known symplectic differential geometry, and examples of such structures related to k•symmetric differential geometry are given.