Solutions of the sine-Gordon equation with a variable amplitude

• Published in: Theoretical and mathematical physics Vol. 184; no. 1; pp. 961 - 972
• Main Authors: Aero, E. L., Bulygin, A. N., Pavlov, Yu. V.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.07.2015
• Subjects: Applications of Mathematics; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical; ansatz; sine-Gordon equation; functionally invariant solution; eikonal equation; wave equation
• ISSN: 0040-5779; 1573-9333
• DOI: 10.1007/s11232-015-0309-8

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.