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Lie symmetry structure of nonlinear wave equations in $(n+1)$-dimensional space-time

P. Basarab-Horwath, F. Güngör, C. Özemir

Abstract

We study Lie point symmetry structure of generalized nonlinear wave equations in the $(n+1)$-dimensional space-time.

Lie symmetry structure of nonlinear wave equations in $(n+1)$-dimensional space-time

Abstract

We study Lie point symmetry structure of generalized nonlinear wave equations in the -dimensional space-time.

Paper Structure

This paper contains 8 sections, 9 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Equivalence transformations
  4. The symmetry condition
  5. Canonical forms for symmetry vector fields
  6. Symmetries of the form $\eta(x,u)\partial_u$
  7. Preliminary ideas on abelian algebras
  8. Appendix

Key Result

Proposition 3.1

The equivalence group of Eq. waveqn1 is given by transformations together with reducedtransformed, where $U_u\neq 0$ and the transformations $(X^0,X^1,\dots, X^n)\in \mathsf{C}(1,n)$, the conformal group in $(n+1)$-space-time, that is they satisfy with $g^{\mu\nu}=\mathop{\mathrm{diag}}\nolimits (1, -1,\dots, -1)$ and $\lambda(x)>0$ is a smooth function given by

Theorems & Definitions (12)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Example 1
  • Example 2
  • Example 3
  • Theorem 5.1
  • Theorem 5.2
  • Proposition 6.1
  • ...and 2 more