一类非线性波方程尖波解及其动力学性质的分析

CUSP WAVE SOLUTIONS AND DYNAMICAL BEHAVIORS IN A CLASS OF NONLINEAR EQUATION

应用动力系统分岔理论和定性理论研究了一类非线性DegasperisProcesi方程的行波解及其动力学性质,并结合可积系统的特点,利用哈密尔顿系统的能量特征,通过Maple软件绘出其相轨图,再根据行波与相轨道间的对应关系,揭示了不同类型的行波解间的转变与参数变化的关系,并且给出了不同行波间相互转换的参数分岔值,从根本上解释了Peakon产生的原因.数值模拟验证了该方法的正确性.最后给出了相应行波解的表达式.

The dynamical behavior and traveling wave solutions of Degasperis-Procesi equation were studied by using bifurcation theory of dynamical system and qualitative theory. Based on the character of integral system and exploiting the energy of Hamiltonian system, the phase portraits were plotted by using the Maple. The relation between the parameter and the type of different traveling wave solutions was revealed, the bifurcation values of different traveling wave solution were given, and the reason for peakon was shown. Under different parameter conditions, exact traveling wave solutions were also given. The numerical simulation shows the correctness of the theoretical analysis.