A new approach to solution of generalized (3+1)-dimensional sine-Gordon equation

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