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

• URL: http://arxiv.org/abs/2305.11180v1
• Date: Tue, 16 May 2023 10:22:47 GMT
• Title: Conservation Laws for the Nonlinear Klein-Gordon Equation in (1+1)-, (2+1), and (3+1)-dimensions
• Authors: Muhammad Al-Zafar Khan
• Abstract 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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