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Framework for nonlocally related partial differential equation systems and nonlocal symmetries: Extension, simplification, and examples

### Abstract

Any partial differential equation(PDE) system can be effectively analyzed through consideration of its tree of nonlocally related systems. If a given PDE system has local conservation laws, then each conservation law yields potential equations and a corresponding nonlocally related potential system. Moreover, from these conservation laws, one can directly construct independent nonlocally related systems by considering these potential systems individually ( singlets), in pairs , taken all together (one -plet). In turn, any one of these systems could lead to the discovery of new nonlocal symmetries and/or nonlocal conservation laws of the given PDE system. Moreover, such nonlocal conservation laws could yield further nonlocally related systems. A theorem is proved that simplifies this framework to find such extended trees by eliminating redundant systems. The planar gas dynamicsequations and nonlinear telegraph equations are used as illustrative examples. Many new local and nonlocal conservation laws and nonlocal symmetries are found for these systems. In particular, our examples illustrate that a local symmetry of a -plet is not always a local symmetry of its “completed” -plet . A new analytical solution, arising as an invariant solution for a potential Lagrange system, is constructed for a generalized polytropic gas.

© 2006 American Institute of Physics

Received 05 July 2006
Accepted 15 August 2006
Published online 14 November 2006

Acknowledgments:
The authors acknowledge financial support from the National Sciences and Engineering Research Council of Canada and also the second author (A.F.C.) is thankful for support from the Killam Foundation. Research of the last author (N.M.I.) was partially supported by a grant of the President of Ukraine for young scientists (Project No. GP/F11/0061).

Article outline:

I. INTRODUCTION
II. CONSTRUCTION OF CONSERVATION LAWS AND NONLOCALLY RELATED PDE SYSTEMS
A. Direct construction method for finding conservation laws
B. Construction of nonlocally related systems from local conservation laws
C. Conservation laws, nonlocally related PDE systems and nonlocal symmetry analysis of planar gas dynamicsequations
1. Conservation laws and nonlocally related systems
2. Nonlocal symmetry analysis for polytropic gas flows
3. Further conservation laws for a general constitutive function
III. LINEAR DEPENDENCE OF CONSERVATION LAWS AND LOCAL EQUIVALENCE OF POTENTIAL SYSTEMS
A. Linear dependence of conservation laws and tree simplification. Two-dimensional case
B. Linear dependence of conservation laws and tree simplification. General case: independent variables
IV. EXTENDED TREES OF NONLOCALLY RELATED PDE SYSTEMS, NONLOCAL SYMMETRIES AND NONLOCAL CONSERVATION LAWS FOR NONLINEAR TELEGRAPH EQUATIONS
A. Local conservation laws for the NLT equation
B. Point and nonlocal symmetry analysis of NLT equations with power nonlinearities
C. New nonlocal conservation laws for NLT equations with power nonlinearities
V. NONLOCAL SYMMETRY CLASSIFICATION FOR GENERALIZED POLYTROPIC GAS FLOWS
A. Classification of point and nonlocal symmetries
B. Nonlocally related systems and invariant solutions
1. Construction of invariant solutions for generalized polytropic PGD equations
2. An invariant solution from a nonlocal symmetry
VI. CONCLUDING REMARKS

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2006-11-14

2016-02-10

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