Solutions of the sine-Gordon equation with a variable amplitude

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• 作者：E. L. Aero ; A. N. Bulygin ; Yu. V. Pavlov
• 关键词：sine ; Gordon equation ; wave equation ; eikonal equation ; functionally invariant solution ; ansatz
• 刊名：Theoretical and Mathematical Physics
• 出版年：2015
• 出版时间：July 2015
• 年：2015
• 卷：184
• 期：1
• 页码：961-972
• 全文大小：1,388 KB
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• 作者单位：E. L. Aero (1)
  A. N. Bulygin (1)
  Yu. V. Pavlov (1)
  
  1. Institute of Problems in Mechanical Engineering, RAS, St. Petersburg, Russia
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Applications of Mathematics
  Russian Library of Science
  
• 出版者：Springer New York
• ISSN：1573-9333

文摘

We propose methods for constructing functionally invariant solutions u(x, y, z, t) of the sine-Gordon equation with a variable amplitude in 3+1 dimensions. We find solutions u(x, y, z, t) in the form of arbitrary functions depending on either one (α(x, y, z, t)) or two (α(x, y, z, t), β(x, y, z, t)) specially constructed functions. Solutions f(α) and f(α, β) relate to the class of functionally invariant solutions, and the functions α(x, y, z, t) and β(x, y, z, t) are called the ansatzes. The ansatzes (α, β) are defined as the roots of either algebraic or mixed (algebraic and first-order partial differential) equations. The equations defining the ansatzes also contain arbitrary functions depending on (α, β). The proposed methods allow finding u(x, y, z, t) for a particular, but wide, class of both regular and singular amplitudes and can be easily generalized to the case of a space with any number of dimensions. Keywords sine-Gordon equation wave equation eikonal equation functionally invariant solution ansatz