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 v = e^iu. 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⁡(μξ) when p = 1 and tanh⁡(μξ/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.