New approach to the solution of the classical sine-Gordon equation and its generalizations

• Published in: Differential equations Vol. 47; no. 10; pp. 1442 - 1452
• Main Authors: Aero, E. L., Bulygin, A. N., Pavlov, Yu. V.
• Format: Journal Article
• Language: English
• Published: Dordrecht SP MAIK Nauka/Interperiodica 01.10.2011; Springer Nature B.V
• Subjects: Algebra; Amplitudes; Derivatives; Difference and Functional Equations; Differential equations; Exact solutions; Mathematical analysis; Mathematics; Mathematics and Statistics; Methods; Nonlinear equations; Ordinary Differential Equations; Partial Differential Equations; Studies; Three dimensional; Ultrashort Optical Pulse; Arbitrary Function; Jacobi Equation; Inverse Scattering; Gordon Equation
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S0012266111100077

We obtain new exact solutions U ( x, y, z, t ) of the three-dimensional sine-Gordon equation. The three-dimensional solutions depend on an arbitrary function F ( α ) whose argument is a function α ( x, y, z, t ). The ansatz α is found from an equation linear in ( x, y, z, t ) whose coefficients are arbitrary functions of α that should satisfy a system of algebraic equations. By this method, we solve the classical and a generalized sine-Gordon equation; the latter additionally contains first derivatives with respect to ( x, y, z, t ). We separately consider an equation that contains only the first derivative with respect to time. We present approaches to the solution of the sine-Gordon equation with variable amplitude. The considered methods for solving the sine-Gordon equation admit a natural generalization to the case of integration of the same types of equations in a space of arbitrarily many dimensions.