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Noether and partial Noether approach for the nonlinear (3+1)-dimensional elastic wave equations.

Akhtar Hussain1, M Usman2, Fiazuddin Zaman1

  • 1Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan.

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Summary

This study uses the Lie group method to find analytical solutions for nonlinear elastic wave equations. Novel approaches were developed to uncover conservation laws for both standard and damped versions of these equations.

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Area of Science:

  • Applied Mathematics
  • Theoretical Physics
  • Nonlinear Dynamics

Background:

  • Nonlinear differential equations model complex physical phenomena.
  • Analytical solutions are crucial for understanding wave propagation.
  • Lie group methods offer powerful tools for solving such equations.

Purpose of the Study:

  • To investigate nonlinear elastic wave equations using Lie group invariants.
  • To derive and analyze conservation laws for both undamped and damped wave equations.
  • To extend the application of variational calculus to these models.

Main Methods:

  • Lie group method for symmetry analysis and deriving group-invariant solutions.
  • Noether's theorem for conservation laws in the presence of a classical Lagrangian.
  • Extended approach using partial Lagrangians for damped equations lacking a classical Lagrangian.

Main Results:

  • An eight-dimensional symmetry algebra was derived for the (3+1)-dimensional nonlinear elastic wave equation.
  • Optimal systems and group-invariant solutions were successfully obtained.
  • Conservation laws for linear momentum and energy were identified for both equation types.

Conclusions:

  • The Lie group method provides effective analytical solutions for nonlinear elastic wave equations.
  • Novel techniques were established for deriving conservation laws in damped systems.
  • This research expands the application of variational calculus in nonlinear wave analysis.