Multi soliton solution for the system of Coupled Korteweg-de Vries equations

A Quadratic B-Spline Galerkin Approach for Solving a Coupled KDV Equation

“…As a final problem, we consider the coupled KdV equation given by (1.1) with the initial conditions U (x, 0) = e −0.01x 2 , V (x, 0) = e −0.01x 2 and the boundary conditions (1.2). All calculations for this problem are done in the range −50 ≤ x ≤ 150 for the values a = 0.5 and b = −3.…”

“…Siraj-ul-Islam et al [15] have formulated a simple classical radial basis functions collocation method for the numerical solution of the coupled KdV equations. Rady et al [2] have considered the system of coupled KdV equations and established the transformation which turns the coupled KdV equations into the single nonlinear partial differential equation, then they obtained an auto-Backlund transformation and lax pairs using the extended homogeneous balance method. Biswas and Ismail [6] have used solitary wave ansatz to carry out the integration of the coupled KdV equation with power law nonlinearity, and then supplemented their results by numerical simulations.…”

The investigation of dynamical behaviour variation of (2+1)-dimensional Extended Calogero-Bogoyavlenskii-Schiff Equation with different fractional operators

“…But now the interest is increasing to study localized soliton solutions in higher dimensions. Numerous techniques, including the homogenous balance method [13][14][15][16][17][18], the hyperbolic function expansion method [21,22], the sine-cosine method [23], the nonlinear transformation method [24][25][26], and the trial function method [27,28], have been proposed in order to obtain the periodic wave and soliton solutions of the nonlinear evolution equations. The generalised periodic solutions cannot be derived using these approaches, which can only produce shock or solitary wave solutions, or periodic wave solutions in terms of basic functions.…”

Variational approach, soliton solutions and singular solitons for new coupled ZK system

“…So we should search for a mathematical algorithm to discover the exact solutions of nonlinear partial differential equations. In recent years, powerful and efficient methods explored to find analytic solutions of nonlinear equations have drawn a lot of interest by a variety of scientists, such as Adomian decomposition method [2], the homotopy perturbation method [3,4], some new asymptotic methods searching for solitary solutions of nonlinear differential equations, nonlinear differential-difference equations and nonlinear fractional differential equations using the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform and ancient Chinese mathematics [4], the variational iteration method [5,6] which is used to introduce the definition of fractional derivatives [7,4], the He's variational approach [8], the extended homoclinic test approach [9,10], homogeneous balance method [11][12][13][14], Jacobi elliptic function method [15][16][17][18], Băclund transformation [19,20], G ′ /G expansion method for nonlinear partial differential equation [21,22], and fractional differential-difference equations of rational type [23][24][25] It is important to point out that a new constrained variational principle for heat conduction is obtained recently by the semi-inverse method combined with separation of variables [26], which is exactly the same with He-Lee's variational principle [27]. A short remark on the history of the semi-inverse method for establishment of a generalized variational principle is given in [28].…”