Effects of Quadratic Singular Curves in Integrable Equations
Effects of Quadratic Singular Curves in Integrable Equations
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DOI：
10.1111/sapm.12060
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发表时间：
2015-01
影响因子：
2.7
通讯作者：
Aiyong Chen;Shu Wen;Shengqiang Tang;Wentao Huang;Z. Qiao
中科院分区：
文献类型：
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作者：
Aiyong Chen;Shu Wen;Shengqiang Tang;Wentao Huang;Z. Qiao
本文利用动力系统的分支理论研究了可积波动方程中二次奇异曲线的作用。一些新的奇异孤立波（pseudo-cuspons）和周期波被发现比规则的奇异行波，如尖峰孤子（peakon），尖点孤子（cuspon），尖点周期波等更弱。我们表明，当新的奇异孤立波和周期波的一阶导数存在时，它们的二阶导数对于孤立波在有限个点处是不连续的，或者对于周期波在无穷个点处是不连续的。此外，在行波系统的相平面上，奇异行波与二次奇异曲线之间建立了一种内在联系。如果相应的周期轨道或同宿轨道与双曲线、椭圆和抛物线相切，则会出现新的奇异周期波、伪尖点和双曲点。特别地，首次提出了伪尖点。最后，我们通过理论分析和数值模拟研究了新的奇异孤立波和周期波解的定性行为。
In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo‐cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first‐order derivatives of the new singular solitary wave and periodic waves exist, their second‐order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo‐cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo‐cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.