New conservation laws obtained directly from symmetry action on a known conservation law

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Abstract

Two formulas are introduced to directly obtain new conservation laws for any system of partial differential equations from a known conservation law and admitted symmetries. The first formula maps any conservation law of a given system to the corresponding conservation law of the system obtained through a contact transformation. When the contact transformation is a symmetry of the given system, then the corresponding conservation law is a conservation law of the given system. The second formula checks a priori whether or not the action of a symmetry (continuous or discrete) on a conservation law can yield one or more new conservation laws of the given system. Several examples are considered, including the use of a discrete symmetry to obtain a new conservation law and the use of a continuous symmetry to generate two new conservation laws.

Keywords

Conservation laws
Symmetry
Nonlinear telegraph equations
Cylindrical KdV equation

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The authors acknowledge financial support from the National Sciences and Engineering Research Council of Canada and also the second author is thankful for partial support from NSFC (No. 10461005).

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Permanent address: Inner Mongolia University of Technology, Hohhot, 010062, China.