Journal ArticleDOI
An operator approach to symmetries
TLDR
In this paper, a single general operator symmetry condition is derived by using the methods of nonlinear functional analysis and a simple and systematic procedure for finding symmetry conditions in particular problems is provided.Abstract:
By using the methods of nonlinear functional analysis a single general operator symmetry condition is derived. It supplies a simple and systematic procedure for finding symmetry conditions in particular problems. From it the Lie algebra structure associated with symmetries is easily displayed and the classical Lie theory of symmetries of differential equations is obtained without needing the technique of the group extension.read more
Citations
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A Simple model of the integrable Hamiltonian equation
TL;DR: In this paper, a method of analysis of the infinite-dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested, based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equation with their symmetries by using symplectic operators.
Journal ArticleDOI
A symmetry approach to exactly solvable evolution equations
TL;DR: In this article, a method is developed for establishing the exact solvability of nonlinear evolution equations in one space dimension which are linear with constant coefficient in the highest order derivative.
Journal ArticleDOI
Equivalence transformations for nonlinear evolution equations
TL;DR: In this paper, the Hopf-Cole transformation for the Burgers equation and the Miura, Backlund, and Hirota transformations for the Korteweg-de-Vries equation are all deduced from a single general equivalence condition.
References
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Book
Similarity methods for differential equations
George W. Bluman,Julian D. Cole +1 more
TL;DR: In this paper, the authors define the notion of groups of transformations and prove that a one-parameter group essentially contains only one infinitesimal transformation and is determined by it.
Journal ArticleDOI
The conceptual basis and use of the geometric invariance principles
TL;DR: In this paper, the authors present an attempt at a formulation of geometric invariances which can be applied to the classical mechanics of point particles, to a relativistic mechanics of interacting point particles and to quantum theory.
Journal ArticleDOI
An operator approach to Poisson brackets
TL;DR: In this article, a general approach to Poisson brackets, based on the study of the Lie algebra of potential operators with respect to closed skew-symmetric bilinear forms, is proposed.
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