Soliton Solutions of Coupled KdV System from Hirota's Bilinear Direct Method

Abstract

With Hirota's bilinear direct method, we study the special coupled KdV system to obtain its new soliton solutions. Then we further discuss soliton evolution, corresponding structures, and interesting interactive phenomena in detail with plot. As a result, we find that after the interaction, the solitons make elastic collision and there are no exchanges of their physical quantities including energy, velocity and shape except the phase shift.

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

“…where a i (i = 0, 1, 2) and b i (i = 0, 1, 2) are constants which are to be determined later, F(ξ) is a Jacobi's elliptic solution of ξ, which satisfies F 2 (ξ) = q 0 + q 2 F ξ) 2 + q 4 F ξ) 4 , and ξ = ω t 0 δ(τ)dτ + λx, where λ(λ = 0) is an arbitrary constant. Substituting (3), ( 4) with ( 1), (2), we have…”

“…We then illustrate the Jacobi's elliptic function solutions of the coupled KdV equations with variable coefficients. For proper values of the parameters q 0 , q 2 and q 4 , the ordinary differential equation 4 can be easily solved, and the corresponding Jacobi's elliptic function solutions are summarized in Table 1. Equations ( 19) and (20) admit 12 different kinds of Jacobi's elliptic functions.…”

“…To date, extensive research has been carried out on the complex nonlinear waves of KdV equations. Many methods have been developed to solve these equations, such as inverse scattering transformation [1], Darboux-Bäcklund transformation [2,3], Hirota's bilinear method [4,5], the first integral method [6], the homogeneous balance principle [7,8], the F-expansion method [9][10][11], the Wronskian method [12], the variational method [13], and Painlevé analysis [14]. On the other hand, KdV equations have different forms in different models, and the relevant methods were developed to study different types of KdV equations.…”