Approximate Damped Oscillatory Solutions for Generalized KdV-Burgers Equation and Their Error Estimates

and Applied Analysis 3 25 , the traveling wave solution of 1.2 is a damped oscillatory solution which has a bell profile head. However, they did not present any analytic solution or approximate solution for 1.2 . Xiong 25 obtained a kink profile solitary wave solution for KdV-Burgers equation. In 26 , S. D. Liu and S. D. Liu obtained an approximate damped oscillatory solution to a saddle-focus kink profile solitary solution of KdV-Burgers equation, without giving its error estimate. However, it is very important to give its error estimate, or people will feel unreliable. From the above references, we can see that there is not only theoretical but also practical significance to find damped oscillatory solutions. However, the above references did not give exact or approximate damped oscillatory solutions of 1.1 when p ≥ 1, and we have not seen the references about it. In this paper, we focus on studying the relations between the behaviors of bounded traveling wave solutions and the dissipation coefficient, showing the reason why the damped oscillatory solutions take place and how to obtain the approximate damped oscillatory solutions and giving their error estimates. We will obtain all the results in 1, Theorem 1.1 when p is natural number, as well as the existent number of bounded traveling wave solutions and relations between the behaviors of bounded traveling wave solutions and the dissipation coefficient α in the case of b > 0, c < 0, b < 0, c > 0, and b < 0, c < 0, respectively. More importantly, we will give approximate damped oscillatory solution and its error estimate in the case of c > 0, 0 < α < 2√pc when p is any natural number. This paper is organized as follows. In Section 2, we carry out qualitative analysis for the planar dynamical system corresponding to 1.1 . We present all global phase portraits of this planar dynamical system and give the existent conditions and number of bounded traveling wave solutions of 1.1 . We obtain that, if α 0, 1.1 at most has two bell profile solitary wave solutions or two kink profile solitary wave solutions and that, if α/ 0, 1.1 at most has two bounded traveling wave solutions kink profile or oscillatory traveling wave solutions . In Section 3, we discuss the relations between the behaviors of bounded traveling wave solutions and the dissipation coefficient α. We find out two critical values λ1 2 √ pc and λ2 2 √−c and obtain that for the righttraveling wave of the equation, a bounded traveling wave solution appears as a monotones kink profile solitary wave solution if dissipation coefficient α ≥ λ1, while it appears as a damped oscillatory wave if 0 < α < λ1; for the left-traveling wave of the equation, a bounded traveling wave solution appears as a monotones kink profile solitary wave solution if dissipation coefficient α ≥ λ2, while it appears as a damped oscillatory wave if 0 < α < λ2. In Section 4, the exact bell profile and kink profile solitary wave solutions of 1.1 without dissipation effect are presented. Furthermore, according to the evolution relation of solution orbits in global phase portraits, by undetermined coefficients method, we obtain approximate damped oscillatory solutions of 1.1 . In Section 5, we study the error estimate between approximate damped oscillatory solutions and their exact solutions. The difficulty of this problem is that we only know the approximate damped oscillatory solutions, but do not know their exact solutions. To overcome it, we use some transformations and the idea of homogenization principle and then establish the integral equations reflecting the relations between the exact solutions and approximate damped oscillatory solutions. Thus, we give error estimates for the approximate solutions obtained in Section 4. We can see that the errors between the exact solutions and approximate damped oscillatory solutions we obtained by this method are infinitesimal decreasing in exponential form. 4 Abstract and Applied Analysis 2. Qualitative Analysis to Bounded Traveling Wave Solutions of 1.1 Assume that u x, t u ξ u x − ct is a traveling wave solution of 1.1 , and u ξ satisfies −cu′ ξ bu ξ u′ ξ − αu′′ ξ u′′′ ξ 0, 2.1 where c is the wave speed. Integrating the above equation once yields u′′ ξ − αu′ ξ − cu ξ b p 1 u 1 ξ g, 2.2 where g is an integral constant. Owing that we focus on studying dissipation effect to the system, we assume that the traveling wave solutions we study satisfy u′ ξ , u′′ ξ −→ 0, |ξ| −→ ∞ 2.3 and the asymptotic values C and C− C limξ→ ∞u ξ , C− limξ→−∞ ξ satisfy b p 1 x 1 − cx 0, 2.4 so under the hypothesis 2.3 and 2.4 , the traveling wave solutions of 1.1 satisfy u′′ ξ − αu′ ξ − cu ξ b p 1 u 1 ξ 0 2.5 Remark 2.1. In the following discussion, we will always assume that the traveling wave solutions of 1.1 satisfy 2.3 and 2.4 . Letting x u ξ and y u′ ξ , then 2.5 can be reformulated as a planar dynamical system