New approach to the solution of the classical sine-Gordon equation and its generalizations (Q662455)

New exact solutions of the three-dimensional sine-Gordon (SG) equation are obtained. These solutions depend on arbitrary function \(F(\alpha)\), which argument is some function \(\alpha(x, y, z, t)\). The ansatz \(\alpha\) is found from the linear equation with respect to \(x, y, z, t,\) whose coefficients are arbitrary functions depending on \(\alpha\). These coefficients must satisfy a system of algebraic equations. By this method, the classical and generalized SG-equations with first derivatives with respect to \(x, y, z, t\) are solved. The SG-equation with only first time derivative is considered separately. Approaches for solutions of SG-equation with variable amplitude are proposed. These methods admit natural generalization in case of integration of the above mentioned types of equations in a space with any number of dimensions.