Solving the forced generalized sinh-Gordon equation using the extended generalized tanh-like ansatz method

Abstract

The generalized sinh-Gordon equation (shG), mechanically represented by a combination of nonlinear springs, is a lesser- known cousin of the sine-Gordon equation (SG). This partial differential equation (PDE) has been solved using the tanh method after the transformation v = e^βu. However, if we use the transformation u = (4/β) artanh(v) based on the standard 1-soliton solution, the tanh method results in a solution with an imaginary factor. Further exploration of this complex solution suggests the use of the recently proposed Generalized Half-Angle Tangent Hyperbolic (g-HATH) ansatz defined as Y_p = [(1+|p|) tanh⁡(μξ/2)]/[1+p tanh² (μξ/2)] in the negative p regime. It simplifies to sinh⁡(μξ) when p = −1 and tanh⁡(μξ/2) when p = 0. The method is composed of three steps. First, the appropriate transformations are applied to shG to convert it into polynomial form. Second, we substitute g-HATH and solve the obtained system of equations. Lastly, the necessary external driving function f(x,t) for the parameterized ansatz is derived to achieve the complete solution. Our approach discovers complex solutions to a classical system mediated by f(x,t), which is controlled by the single parameter p.