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A new conservation theorem. (English) Zbl 1160.35008
A classical result in the calculus of variations is the Noether theorem that every variational symmetry has an associated conservation law. The author proposes an extension of this result to arbitrary differential equations not necessarily of a variational nature. It is based on a novel concept of adjoint for nonlinear equations. The author shows first that the system consisting of the original equation plus its adjoint is variational and then that every symmetry of the original equation can be extended to one of the combined system. Now a straightforward application of the classical Noether theorem yields a conservation law. As the definition of a nonlinear adjoint requires the introduction of additional unknown functions, one obtains in general even an infinite familiy of conservation laws parametrised by solutions of the adjoint equation. As concrete examples the heat and the Korteweg-de Vries equations are studied.

35A30Geometric theory for PDE, characteristics, transformations
58J70Invariance and symmetry properties
Full Text: DOI
[1] Ibragimov, N. H.: Transformation groups in mathematical physics. (1983) · Zbl 0529.53014
[2] Ibragimov, N. H.: Elementary Lie group analysis and ordinary differential equations. (1999) · Zbl 1047.34001
[3] Ibragimov, N. H.: Integrating factors, adjoint equations and Lagrangians. J. math. Anal. appl. 318, No. 2, 742-757 (2006) · Zbl 1102.34002
[4] Ibragimov, N. H.; Shabat, A. B.: Korteweg -- de Vries equation from the group-theoretic point of view. Dokl. akad. Nauk SSSR 244, No. 1, 57-61 (1979) · Zbl 0423.35076
[5] Noether, E.: Invariante variationsprobleme. Königliche gesellschaft der wissenschaften zu göttingen, nachrichten. Mathematisch-physikalische klasse heft 2, No. 3, 235-257 (1918)