AI 中文总结
本研究为奇异摄动KdV方程构造带相移的Nanopteron解，明确振幅与相移求解差异，其简单结构可作为更复杂问题Nanopteron构造的模板。
AI 中文摘要
我们为微分方程ε²u⁽⁴⁾+u''−u+u²=0构造Nanopteron解，该方程是Korteweg–de Vries（KdV）方程行波问题u''−u+u²=0的奇异摄动形式。这些Nanopteron解是sech²型KdV行波剖面、小局部误差与周期性“波纹”的叠加，波纹由振幅和相移参数化。构造Nanopteron时，我们预选其中一个参数再求解另一个，并详细说明振幅与相移求解过程的差异。结合格微分方程Nanopteron的现代进展，本研究重新审视了该奇异摄动KdV方程Nanopteron解的若干已有构造，其简单结构可作为更复杂问题中Nanopteron构造的易获取模板。
英文摘要
We construct nanopteron solutions to the differential equation \[ ε^2u^{(4)}+u''-u+u^2 =0, \] which is a singular perturbation of the traveling wave problem $u''-u+u^2=0$ for the Korteweg--de Vries (KdV) equation. These nanopteron solutions are the superposition of the sech$^2$-type KdV traveling wave profile, a small localized error, and a periodic "ripple." The ripple is parametrized by its amplitude and its phase shift. In the process of constructing the nanopteron, we preselect one of these parameters and then solve for the other, and we provide detailed commentary on how the amplitude and phase shift solution processes differ. Our work revisits several prior constructions of nanopteron solutions for this singularly perturbed KdV equation in light of modern developments of nanopterons for lattice differential equations. In the context of these contemporary methods, the simple structure of the singularly perturbed KdV equation allows it to serve as an accessible template for nanopteron constructions in more complicated problems.
