On the identification of nonlinear terms in the generalized Camassa-Holm equation involving dual-power law nonlinearities

Abstract

© 2020 IMACS The nonlinear waves constitute a significant concern in the design in order to understand wave behaviors and structures. Due to the complexity of the waves phenomenon, many mathematical models have been coming up and extensively being studied until now. In this paper, we introduce a new aspect to study a wave model by coupling of the classical Camassa-Holm equation and the BBM-KdV equation with the dual-power law nonlinearities. The new model still satisfies the fundamental energy conservative property as the original models. As we have known, the use of an efficient and accurate numerical method can be a beneficial tool for studying the wave behaviors; therefore, we aim to propose a finite difference scheme for solving the new model. A three-level finite difference technique is applied here in order to be able to design a linear scheme. The highlight of our scheme is to propose an alternative way to discretize the highly nonlinear terms. The class of analytical solutions of the model is derived and used to validate our numerical scheme. Like the design, the energy conservation law is preserved through the numerical scheme. In the context of theoretical analysis, the existence and uniqueness is analyzed based on a priori estimate. The stability and convergence of the numerical solution with second-order accuracy on both space and time are also provided under a mild restriction on the ratio of τ/h. Many comparisons of our scheme and a modification of the existing finite difference scheme are preformed, and the consequences confirm that the proposed scheme gives a significant improvement over the other. Moreover, in the numerical simulations, the faithfulness of the proposed method is validated by the pieces of evidence of depression solitary waves under the effect of the power of nonlinearities.