Bright Soliton Behaviours of Fractal Fractional Nonlinear Good Boussinesq Equation with Nonsingular Kernels

Abstract

In this manuscript, we investigate the nonlinear Boussinesq equation (BEQ) under fractal-fractional derivatives in the sense of the Caputo–Fabrizio and Atangana–Baleanu operators. We use the double modified Laplace transform (LT) method to determine the general series solution of the Boussinesq equation. We study the convergence, existence, uniqueness, boundedness, and stability of the solution of the considered good BEQ under the aforementioned derivatives. The obtained solutions are presented with numerical illustrations considering a particular example by two cases based on both derivatives with suitable initial conditions. The results are illustrated graphically where good agreements are obtained. Our results show that fractal-fractional derivatives are a very effective tool for studying nonlinear systems. Furthermore, when t increases, the solitary waves of the system oscillate. As the fractional order α or fractal dimension β increases, the soliton solutions become coherently close to the exact solution. For compactness, an error analysis is performed. The absolute error reveals an approximate linear evolution in the soliton solutions as time increases and that the system does not blow up nonlinearly.