一类扰动的Degasperis-Procesi方程行波解的存在性研究

Study on the Existence of Traveling Wave Solutions for a Class of Perturbed Degasperis-Procesi Equations

摘要: 本文研究了一类扰动Degasperis-Procesi (DP)方程行波解的存在性。利用Melnikov函数方法和几何奇异摄动理论，分析了方程$u_t-u_{xxt}+k{u}_x+\alpha u{u}_x+ϵ({\lambda }_1{u}_{xx}+{\lambda }_2{u}_{xxtx}+{\lambda }_3{\left(u{u}_x\right)}_x+{\lambda }_4{\left(u_x{u}_{xx}\right)}_x+{\lambda }_2{\left(u{u}_{xxx}\right)}_x$ $+{\lambda }_5{\left(u^2{u}_x\right)}_x)=3u_x{u}_{xx}+u_x{u}_{xxx}$ 行波解的存在性。通过行波变换，将该偏微分方程转化为扰动的常微分系统来进行探究。当$k=c-\frac{\alpha c}{2}$ ，通过分析其对应的常微分系统，证明了无扰动DP方程存在孤立波解。此外，通过考虑对应的扰动的常微分系统的极限环个数，运用Melnikov函数方法，证明了该方程存在2个孤立的周期波解。

Abstract: This paper investigates the existence of traveling wave solutions for a class of perturbed Degasperis-Procesi equations. Utilizing the Melnikov function method and geometric singular perturbation theory, the equation$u_t-u_{xxt}+k{u}_x+\alpha u{u}_x+ϵ({\lambda }_1{u}_{xx}+{\lambda }_2{u}_{xxtx}+{\lambda }_3{\left(u{u}_x\right)}_x+{\lambda }_4{\left(u_x{u}_{xx}\right)}_x$ $+{\lambda }_2{\left(u{u}_{xxx}\right)}_x+{\lambda }_5{\left(u^2{u}_x\right)}_x)=3u_x{u}_{xx}+u_x{u}_{xxx}$ . is analyzed for traveling wave solutions. Through a traveling wave transformation, the partial differential equation (PDE) is converted into a perturbed ordinary differential system for investigation. When $k=c-\frac{ac}{2}$ , by analyzing the corresponding ordinary differential system, it is proven that the unperturbed DP equation has solitary wave solutions. Moreover, by considering the number of limit cycles of the corresponding perturbed ordinary differential system and applying the Melnikov function method, it is shown that there exist two isolated periodic wave solutions for the equation.