A Generalized KdV Equation of Neglecting the Highest-Order Infinitesimal Term and Its Exact Traveling Wave Solutions

and Applied Analysis 3 hard to obtain because they are highly nonlinear equations and most probably they are not integrable equations in general. Thus, large numbers of research results are still concentrated in the classical KdV equation and some other high-order equations with KdV type, such as KdV-Burgers equation [17, 18] and KdV-Burgers-Kuramoto equation [19], at present. Therefore, the investigation of the more exact solutions for (1) is very important and necessary. However, by using the current methods, we can not obtain exact solutions of (1) in universal conditions; the next best thing is the investigation of exact solutions of (7). In this paper, still regarding the ρ i (i = 1, 2, 3, 4) as free parameters and by using the integral bifurcation method [20, 21], we will investigate exact traveling wave solutions and their properties of (7). The rest of this paper is organized as follows. In Section 2, wewill derive two-dimensional planar systemwhich is equivalent to (7) and give its first integral equation. In Section 3, by using the integral bifurcationmethod, we will obtain some new travelingwave solutions and discuss their dynamic properties. 2. Two-Dimensional Planar Dynamical System of (7) and Its First Integral and Conservation of Energy Making a transformation η(t, x) = φ(ξ) with ξ = x − ct, (7) can be reduced to the following ODE: − cφ 󸀠 + φ 󸀠 + αφφ 󸀠 + βφ 󸀠󸀠󸀠 + ρ 1 α 2 φ 2 φ 󸀠 + αβ (ρ 2 φφ 󸀠󸀠󸀠 + ρ 3 φ 󸀠 φ 󸀠󸀠 ) + ρ 4 α 3 φ 3 φ 󸀠 = 0, (11) where c is wave velocity which moves along the direction of x-axis and c ̸ = 0. Integrating (11) once and setting the integral constant as zero yields (1 − c) φ + 1 2 αφ 2 + βφ 󸀠󸀠 + 1 3 ρ 1 α 2 φ 3 + αβ × [ρ 2 φφ 󸀠󸀠 + 1 2 (ρ 3 − ρ 2 ) (φ 󸀠 ) 2 ] + 1 4 ρ 4 α 3 φ 4 = 0. (12) Let φ󸀠 = y. Thus (12) can be reduced to a planar system