Exact Solutions for Coupled Variable Coefficient KdV Equation via Quadratic Jacobi’s Elliptic Function Expansion

Abstract

The exact traveling wave solutions to coupled KdV equations with variable coefficients are obtained via the use of quadratic Jacobi’s elliptic function expansion. The presented coupled KdV equations have a more general form than those studied in the literature. Nine couples of quadratic Jacobi’s elliptic function solutions are found. Each couple of traveling wave solutions is symmetric in mathematical form. In the limit cases m→1, these periodic solutions degenerate as the corresponding soliton solutions. After the simple parameter substitution, the trigonometric function solutions are also obtained.

Retracted: Exploring new optical solutions for nonlinear Hamiltonian amplitude equation via two integration schemes

“…Jacobi elliptic functions have numerous applications in mathematical physics, including in the theory of elliptic integrals, the study of periodic solutions of nonlinear differential equations, and the analysis of periodic structures in materials science. The Jacobi elliptic function expansion method has been applied to a wide range of nonlinear partial differential equations [29][30][31][32][33][34][35][36][37][38]. The fundamental outline of JEFEM can be summarized as follows:…”

“…expansion method [11,12], the Bernoulli subequation function method [13], the generalized exponential rational function method [14-16], the ¢ -( ) G 1 expansion method [17, 18], Hirota's simple method [19][20][21], and other methods [22][23][24][25][26][27] have been used to obtain solutions for these NPDEs. Additionally, the Jacobi elliptic function expansion method (JEFEM) has been applied to several NPDEs, including the Biswas-Arshed equation in [28], various nonlinear wave equations such as KdV, mKdV equations, Boussinesq model and nonlinear klein-gordon equation [29], the fourth-order NPDE and the Kaup-Newell equation in [30], and other related studies [31][32][33][34][35][36][37][38].Similarly, the new modification of the Sardar sub-equation method (MSSEM) has also been applied to several NPDEs such as the generalized unstable nonlinear Schrodinger equation in [39], the perturbed Fokas-Lenells equation in [40], the Korteweg-de Vries equation in [41], the Benjamin-Bona-Mahony equation and the Klein-Gordon equations in [42], the Klein-Fock-Gordon equation in [43], the Boussinesq equation in [44], the perturbed Gerdjikov-Ivanov equation in [45], and other related studies [46][47][48][49][50][51][52].…”

Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schrödinger Equations

“…In mathematical physics, there are some effective methods [1][2][3][4][5][6][7][8] for the solitary wave solutions of nonlinear evolution equations, such as the inverse scattering method [1], DT [3], Hirota bilinear approach [4], and the exp-function method [6]. The DT [3] shows its algebraic operation characteristics in constructing multi-soliton solutions [9][10][11][12][13][14], which creates favorable conditions for the GDT [15][16][17] to construct semirational solutions with solitrogon structures [18][19][20][21].…”