Adomian Decomposition Method for Approximating the Solutions of the Bidirectional Sawada-Kotera Equation

Abstract

In this paper, the decomposition method is implemented for solving the bidirectional Sawada- Kotera (bSK) equation with two kinds of initial conditions. As a result, the Adomian polynomials have been calculated and the approximate and exact solutions of the bSK equation are obtained by means of Maple, such as solitary wave solutions, doubly-periodic solutions, two-soliton solutions. Moreover, we compare the approximate solution with the exact solution in a table and analyze the absolute error and the relative error. The results reported in this article provide further evidence of the usefulness of the Adomian decomposition method for obtaining solutions of nonlinear problems

Non-local residual symmetry and soliton-cnoidal periodic wave interaction solutions of the KdV6 equation

“…where ω 1 and ω 0 describe the velocities of the soliton and its surrounding W-wave, b 1 and b 0 are two quantities referring to the widths of the soliton and W-wave, respectively. Specially, if W(ξ) = 0 is taken, the solution (14) with Eq. 16 reduces to the trivial traveling wave solution.…”

“…As pointed out in our pervious work [2,7], soliton-cnoidal wave interaction solutions can be regarded as dressed solitons, namely, solitons dressed by cnoidal periodic waves. After taking the limit tanh(f) = ±1 in the solution (14), i.e., removing the soliton cores of v and w, only the cnoidal periodic wave parts remain. To illustrate the soliton-cnoidal periodic wave structure in more detail, it is exhibited in a graphical way with the parameters c 2 = 0, δ = b 1 = λ = 1, ω 0 = 2 and m = 0.01; Figures 1A,B display the profiles of the soliton-cnoidal wave structures at t = 0; Figures 1C,D depict the soliton cores of v and w, where the dashed lines show the left parts of the solitons, i.e., taking tanh(f) = −1 in the solution (14), and the right parts are displayed by the solid lines.…”

“…First, the designable of the velocity of the cnoidal periodic wave is to be considered. Figure 2 exhibits the time-space evolutions of the soliton-cnoidal periodic wave solution (14). The overtaking collision processes between solitons and cnoidal waves are shown in Figure 2A, B.…”

Novel y-type and hybrid solutions for the $$(2+1)$$-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

“…Therefore, when β = 0 (3) reduces to the standard KdV equation [36,37]. While when α = 0 (3) reduces to the (2+1)-dimensional SK equation [38][39][40].…”

Soliton Molecule and Breather-Soliton Molecule Structures for a General Sixth-Order Nonlinear Equation

“…and the Kaup-Kupershmidt (KK) equation [8−10] 5𝜕 −1 𝑥 𝑣 𝑡𝑡 + 5𝑣 𝑥𝑥𝑡 − 15𝑣𝑣 𝑡 − 15𝑣 𝑥 𝜕 −1 𝑥 𝑣 𝑡 − 45𝑣 2 𝑣 𝑥 + 75 2 𝑣 𝑥 𝑣 𝑥𝑥 + 15𝑣 𝑥 𝑣 𝑥𝑥𝑥 − 𝑣 5𝑥 = 0, (7) which describe the propagation of waves in two opposite directions. Equations ( 6) and (7) possess Lax pairs due to Dye and Parker's [4] construction and fall into the class (1) after the potential transformation 𝑣 = 𝑢 𝑥 . The nonlinear wave Eq.…”