Abstract
Finding exact solutions of nonlinear equations plays an important role in nonlinear science, especially in engineering and mathematical physics. In this paper, we employed the complex method to get eight exact solutions of the modified BBM equation for the first time, including two elliptic function solutions, two simply periodic solutions, and four rational function solutions. We used the exp − ϕ z -expansion methods to get fourteen forms of solutions of the modified BBM equation. We also used the sine-cosine method to obtain eight styles’ exact solutions of the modified BBM equation. Only the complex method can obtain elliptic function solutions. We believe that the complex method presented in this paper can be more effectively applied to seek solutions of other nonlinear evolution equations.
Highlights
It is well known that modern natural science is undergoing profound changes
In order to get the exact solution of mBBM equation, we employ complex method to seek for the exact solution of (3), and we use the exp(− φ(z))-expansion method to get some exact solutions and use sine-cosine method to find the exact solutions of modified Benjamin-Bona-Mahony equation
Yuan et al [35, 36, 39, 42] summarized the work of Eremenko et al and introduced the complex method to find the exact solutions of nonlinear evolution equations in mathematical physics for the first time
Summary
It is well known that modern natural science is undergoing profound changes. Nonlinear science runs through the fields of mathematical science, life science, space science, and electrical engineering [1] and has become an important frontier of contemporary scientific research. In 2003, Liu [30] constructed the exact solution of the BBM equation by using the Jacobi elliptic function expansion method. In 2004, Tautusan [31] constructed an exact solitary wave solution of the BBM equation and the mBBM equation by combining auxiliary equations and hyperbolic function assumptions. We used the complex method [35, 36] and the exp(− φ(z))-expansion method [37] and sine-cosine method [38] to find the exact solutions of modified Benjamin-BonaMahony equation. In order to get the exact solution of mBBM equation, we employ complex method to seek for the exact solution of (3), and we use the exp(− φ(z))-expansion method to get some exact solutions and use sine-cosine method to find the exact solutions of modified Benjamin-Bona-Mahony equation
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