Conservation Laws for the Nonlinear Klein-Gordon Equation in (1+1)-, (2+1), and (3+1)-dimensions

We study soliton solutions to the Klein-Gordon equation via Lie symmetries and the travelling-wave ansatz. It is shown, by taking a linear combination of the spatial and temporal Lie point symmetries, that soliton solutions naturally exist, and the resulting field lies in the complex plane. We norma...

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Bibliographic Details
Published inarXiv.org
Main Author Muhammad Al-Zafar Khan
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 16.05.2023
Subjects
Conservation laws
Klein-Gordon equation
Solitary waves
Traveling waves
Trigonometric functions
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Summary:We study soliton solutions to the Klein-Gordon equation via Lie symmetries and the travelling-wave ansatz. It is shown, by taking a linear combination of the spatial and temporal Lie point symmetries, that soliton solutions naturally exist, and the resulting field lies in the complex plane. We normalize the field over a finite spatial interval, and thereafter, specify one of the integration constants in terms of the other. Solutions to a specific type of nonlinear Klein-Gordon equation are studied via the sine-cosine method, and a real soliton wave is obtained. Lastly, the multiplier method is used to construct conservation laws for this particular nonlinear Klein-Gordon equation in (3 + 1)-dimensions.
ISSN:2331-8422