We study Lie point symmetry structure of generalized nonlinear wave equations of the form <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Box u=F(x, u, \nabla u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>□</mo> <mi>u</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Box $</EquationSource> <EquationSource Format="MATHML"><math> <mo>□</mo> </math></EquationSource> </InlineEquation> is the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(n+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimensional space-time wave (or d’Alembert) operator, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$x\in \mathbb {R}^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>). We find the equivalence groups of this class and its subclass where the first order derivatives are absent. We then determine the symmetry group as a special case of the equivalence from the invariance requirement of the nonlinearity F leading to the symmetry condition involving F. As an application we solve this condition for some specific cases of F to build physically important equations like conformally-invariant nonlinear wave and Euler–Poisson–Darboux equation. Canonical forms for allowable symmetries are also studied.

Lie Symmetry Structure of Nonlinear Wave Equations in $(n+1)$-Dimensional Space-time

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发表日期：

2026年2月6日

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Faruk Güngör,
Cihangir Özemir

摘要

We study Lie point symmetry structure of generalized nonlinear wave equations of the form $\Box u=F(x, u, \nabla u)$ $□ u = F (x, u, \nabla u)$ where $\Box $ $□$ is the $(n+1)$ $(n + 1)$ -dimensional space-time wave (or d’Alembert) operator, $x\in \mathbb {R}^{n+1}$ $x \in R^{n + 1}$ ( $n\ge 2$ $n \geq 2$ ). We find the equivalence groups of this class and its subclass where the first order derivatives are absent. We then determine the symmetry group as a special case of the equivalence from the invariance requirement of the nonlinearity F leading to the symmetry condition involving F. As an application we solve this condition for some specific cases of F to build physically important equations like conformally-invariant nonlinear wave and Euler–Poisson–Darboux equation. Canonical forms for allowable symmetries are also studied.

Lie Symmetry Structure of Nonlinear Wave Equations in \((n+1)\)-Dimensional Space-time

摘要