A Note on the Multiplier Approach for Derivation of Conservation Laws for Partial Differential Equations in the Complex Domain
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We study the conservation laws of scalar partial differential equations (PDEs) with two real independent variables in the complex plane. The complex PDE is split into a system of two real coupled or uncoupled PDEs. We invoke the multiplier method for the derivation of conserved quantities for the complex PDEs and their split systems. The approach is applied to both variational and non-variational complex PDEs. The decomposed complex multipliers of the complex PDE yields two real multipliers for the related split system in the real plane. The multipliers of the split system are derived by utilizing the multiplier method. The multipliers of the split system are compared with the multipliers of the complex PDE after decomposition of the complex multipliers. It is demonstrated that the split multipliers of the complex PDE are not in general the same as the multipliers of the decomposed system of real PDEs. They are shown to be identical when all the multipliers of the complex PDE have either pure real or imaginary parts. We moreover look at the number of conserved vectors that arise by a complex split and from the real system by use of the multipliers.
KeywordsMultiplier approach Complex domain Conservation laws
F.M. is grateful to the N.R.F. of South Africa for a research grant which has facilitated this work.
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