Soliton Solutions of Nonlinear and Nonlocal Sine-Gordon Equation Involving Riesz Space Fractional Derivative

In this article, a novel approach comprising modified homotopy analysis method with Fourier transform has been implemented for the approximate solution of fractional sine-Gordon equation (SGE) ${u_{tt}}\; - \;{}^RD_x^\alpha u\; + \;{\rm{sin}}u\; = \;0,$ where $^RD_x^\alpha $ is the Riesz space fractional derivative, 1 ≤ α ≤ 2. For α=2, it becomes classical SGE utt−uxx + sinu=0, and corresponding to α=1, it becomes nonlocal SGE utt−Hu + sinu=0, which arises in the Josephson junction theory, where H is the Hilbert transform. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α=2) and nonlocal SGE (corresponding to α=1). Here, the approximate solution of fractional SGE is derived by using modified homotopy analysis method with the Fourier transform. Then, we analyse the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method.