Geodesic
Solutions of the sine-Gordon equation with a variable amplitude
Teoretičeskaâ i matematičeskaâ fizika, Tome 184 (2015) no. 1, pp. 79-91 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $(\alpha(x,y,z,t))$ or two $(\alpha(x,y,z,t),\beta(x,y,z,t))$ specially constructed functions. Solutions $f(\alpha)$ and $f(\alpha,\beta)$ relate to the class of functionally invariant solutions, and the functions $\alpha(x,y,z,t)$ and $\beta(x,y,z,t)$ are called the ansatzes. The ansatzes $(\alpha,\beta)$ 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 $(\alpha,\beta)$. 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.
Mots-clés : sine-Gordon equation, eikonal equation, ansatz.
Keywords: wave equation, functionally invariant solution 
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E. L. Aero; A. N. Bulygin; Yu. V. Pavlov. Solutions of the sine-Gordon equation with a variable amplitude. Teoretičeskaâ i matematičeskaâ fizika, Tome 184 (2015) no. 1, pp. 79-91. http://geodesic.mathdoc.fr/item/TMF_2015_184_1_a4/

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