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Cotton-type and joint invariants for linear elliptic systems.

A Aslam1, F M Mahomed2

  • 1Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa ; School of Natural Sciences (SNS), National University of Sciences and Technology, Campus H-12, Islamabad 44000, Pakistan.

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This study derives Cotton-type invariants for systems of linear elliptic equations using two methods, finding they are identical. These invariants are crucial for understanding the geometry of these equations.

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Area of Science:

  • Differential Geometry
  • Mathematical Physics

Background:

  • Systems of linear elliptic equations are fundamental in various scientific fields.
  • Understanding their geometric properties requires specialized invariants.
  • Previous methods for deriving such invariants were limited.

Purpose of the Study:

  • To derive Cotton-type invariants for a subclass of two linear elliptic equations.
  • To establish the identity of invariants obtained through different derivation methods.
  • To generalize the derivation of Cotton-type and joint invariants for general systems.

Main Methods:

  • Splitting complex Cotton invariants of a base complex linear elliptic equation.
  • Utilizing Laplace-type invariants of equivalent linear hyperbolic systems.
  • Applying linear complex transformations to independent variables.

Main Results:

  • Cotton-type invariants for a subclass of linear elliptic systems were successfully derived.
  • Invariants obtained by splitting complex Cotton invariants and from Laplace-type invariants were shown to be identical.
  • A general method for obtaining Cotton-type and joint invariants for systems of two linear elliptic equations was established.

Conclusions:

  • The study confirms the equivalence of different approaches to deriving Cotton-type invariants.
  • The findings provide a robust framework for analyzing the geometry of linear elliptic systems.
  • The presented examples validate the theoretical results and their applicability.