论文标题
通过几何方法在扰动的$（1 + 1）$（1 + 1）$（1 + 1）$（1 + 1）$（1 + 1）中的分叉通过几何方法
Bifurcation of the traveling wave solutions in a perturbed $(1 + 1)$-dimensional dispersive long wave equation via a geometric approach
论文作者
论文摘要
选择$κ$（与同质轨道相关的马鞍点的水平纵坐标）作为分叉参数，以扰动的$（1 + 1）$ - 尺寸分散性长波方程来研究波动波解决方案的分叉。 The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter $κ\in (0,1-\frac{\sqrt3}{3})\cup (1+\frac{\sqrt3}{3},2)$, while the kink and anti-kink wave solutions exist at a unique wave speed $ c^*= \ sqrt {15}/3 $ for $κ= 0 $或$κ= 2 $。这些方法基于几何奇异扰动（短）方法，Melnikov方法和不变歧管理论。有趣的是，不仅为扰动的长波方程直接获得了复杂的同型Melnikov积分的显式分析表达，而且直接给出了极限波速的显式分析表达。数值模拟用于验证我们的数学结果。
Choosing $κ$ (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed $(1 + 1)$-dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter $κ\in (0,1-\frac{\sqrt3}{3})\cup (1+\frac{\sqrt3}{3},2)$, while the kink and anti-kink wave solutions exist at a unique wave speed $c^*=\sqrt{15}/3$ for $κ=0$ or $κ=2$. The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.
