Explicit Exact Solutions of the (2+1)-Dimensional Integro-Differential Jaulent–Miodek Evolution Equation Using the Reliable Methods

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.

Further Results about Seeking for the Exact Solutions of the Nonlinear 2 + 1 -Dimensional Jaulent-Miodek Equation

“…In [37], the authors used both methods, namely, the G ′ / G 2 -expansion method and the Jacobi elliptic equation method to calculate and analyze the explicit exact traveling wave solutions of the ð2 + 1Þ-dimensional Jaulent-Miodek equation as given by equation (1). By applying the G ′ /G 2 expansion method to the equation, the solutions were obtained by the author including the triangular hyperbolic function solutions and rational function solutions.…”

“…In 2019, Shehata and Zahran [36] used the Jaulent-Miodek equation to construct the exact traveling wave solutions of parametric equations with different methods. In 2020, Kaewta et al [37] transformed the ð2 + 1Þ-dimensional Jaulent-Miodek equation into a fourth-order partial differential equation and then obtained the exact solution which is simple and reliable.…”

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

“…In this section, we briefly describe the G /G 2 -expansion method, which is discussed in [21][22][23][24][25][26][27]29,63]. Consider the nonlinear conformable partial differential equation of the unknown function u = u(x 1 , x 2 , ..., x n , t) consisting of the independent variables x 1 , x 2 , ..., x n , and t as follows:…”

“…where ω(c, µ, λ) is defined in (27), c is an arbitrary constant and µ, λ are arbitrary integers. Substituting (35) into solution ( 24), along with the ratio G /G 2 , in ( 15)-( 17), depending on the values of µ and λ, we obtain the solution W(ξ).…”

“…In the past couple of decades, many robust, efficient, and powerful methods exist that have been developed for finding exact solutions of NLEEs, including the (G /G, 1/G)-expansion method [11], the enhanced (G /G)-expansion method [12], the exp-function method [13], the Jacobi elliptic equation method [14], the generalized Kudryashov's method [15], the sine-Gordon expansion method [16], the sub-equation method [17], the improved tan(φ/2)-expansion method [18], and the extended direct algebraic method [19,20]. More recently, the (G /G 2 )-expansion method [4,[21][22][23][24][25][26][27][28][29] has attracted a remarkable amount of attention of many researchers who employed the method to construct exact solutions of certain NPDEs. In 2018, Arshed and Sadia [23] used the (G /G 2 )-expansion method to obtain some new traveling wave solutions for the time-fractional Burgers equation, the fractional biological population model, and the space-time fractional Whitham-Broer-Kaup equations.…”