Abstract
In this article, we utilize the G′/G2-expansion method and the Jacobi elliptic equation method to analytically solve the (2 + 1)-dimensional integro-differential Jaulent–Miodek equation for exact solutions. The equation is shortly called the Jaulent–Miodek equation, which was first derived by Jaulent and Miodek and associated with energy-dependent Schrödinger potentials (Jaulent and Miodek, 1976; Jaulent, 1976). The equation is converted into a fourth order partial differential equation using a transformation. After applying a traveling wave transformation to the resulting partial differential equation, we obtain an ordinary differential equation which is the main equation to which the both schemes are applied. As a first step, the two methods give us distinguish systems of algebraic equations. The first method provides exact traveling wave solutions including the logarithmic function solutions of trigonometric functions, hyperbolic functions, and polynomial functions. The second approach provides the Jacobi elliptic function solutions depending upon their modulus values. Some of the obtained solutions are graphically characterized by the distinct physical structures such as singular periodic traveling wave solutions and peakons. A comparison between our results and the ones obtained from the previous literature is given. Obtaining the exact solutions of the equation shows the simplicity, efficiency, and reliability of the used methods, which can be applied to other nonlinear partial differential equations taking place in mathematical physics.
Highlights
Nonlinear partial differential equations (NPDEs) are extensively used to explain complex phenomena in various fields of applied sciences, especially in physics and engineering
Nonlinear evolution equations, which are formulated using NPDEs, describe more than one of dispersion, dissipation, diffusion, reaction, and convection. e investigation of searching solutions for nonlinear evolution equations plays an important role in nonlinear physical science because the solutions can describe various natural phenomena of the problems such as vibrations, solitons, and propagation with a finite speed. ere are two essential types of solutions for NPDEs, which are analytical and exact solutions
The sine-cosine method [21, 22], the International Journal of Mathematics and Mathematical Sciences tanh-coth method [23, 24], the extended sech-tanh method [25], the sine-Gordon expansion method [26,27,28], and the (G′/G)-expansion method [29,30,31] have been recently utilized to find analytical exact solutions of NPDEs as well. e advantage of finding exact solutions of nonlinear partial differential equations (NPDEs) is that they do not give any error terms for the problems which are better than numerical solutions of the problems
Summary
Nonlinear partial differential equations (NPDEs) are extensively used to explain complex phenomena in various fields of applied sciences, especially in physics and engineering. In 2012, the (G′/G)-expansion method was used to construct some new traveling wave solutions including hyperbolic function, trigonometric function, and rational function solutions of the (2 + 1)-dimensional Jaulent–Miodek equation [40]. Zhang et al [42] solved the (2 + 1)-dimensional Jaulent–Miodek equation using the direct symmetry method for the exact solutions including polynomial solutions, Airy function solutions, elliptic periodic solutions, and rational solutions. In 2018, Gu et al [44] derived exact traveling wave solutions of the (2 + 1)dimensional Jaulent–Miodek equation using the complex method. We aim to use the (G′/G2)-expansion method [46,47,48] and the Jacobi elliptic equation method [49,50,51,52] to solve (2) for its exact traveling wave solutions.
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