Solving the nonlinear forced Sine-Gordon equation PDE using the newly reported generalized tanh-like ansatz method

Abstract

The standard or homogeneous Sine-Gordon equation (SG) is a nonlinear extension of the classical wave equation. This Partial Differential Equation (PDE) was solved using the tanh method through the Painlevé transformation. However, the tanh method fails when the tangent half-angle transformation is introduced to SG. To address this problem, we propose applying a tanh-like function, the Generalized Half-Angle Tangent Hyperbolic (called g-HATH), to the SG. It reduces to tanh(με) when p=1  and tanh(με/2) when p=0. This led to new solutions generated by the appropriate forcing function, both of whose properties vary simply with the single parameter p. Our new approach to the Sine-Gordon equation suggests different solutions controlled by an external driving function.