高阶非线性薛定谔方程中的纳米波：离散化的影响
Nanoptera In Higher-Order Nonlinear Schrödinger Equations: Effects Of Discretization
AI总结：
本研究利用指数渐近方法分析奇异摄动Karpman方程中的纳米波，并研究有限差分离散化对其影响，发现离散化改变振荡幅度、周期及分岔参数，且高阶离散化分岔值趋于非零常数。
摘要：
我们考虑非线性薛定谔方程的推广形式，称之为“Karpman方程”，其中包含额外的线性高阶导数。奇异摄动的Karpman方程产生广义孤立波（GSWs），其形式为带有指数小振荡尾部的孤立波。纳米波是GSWs的一种特殊情况，其中这些振荡尾部不衰减。先前关于连续三阶和四阶Karpman方程的研究表明，纳米波出现在特定设置中。我们使用指数渐近技术来识别奇异摄动连续Karpman方程中的行波纳米波。然后，我们通过将有限差分离散化应用于连续Karpman方程并研究行波解，来研究离散化对纳米波的影响。有限差分离散化将连续Karpman方程转化为一个超前-延迟方程，我们使用指数渐近分析对其进行研究。通过比较这些离散Karpman方程中的纳米波与其连续对应物中的纳米波，我们表明纳米波尾部的振荡幅度和周期在连续方程和离散方程中有所不同。我们还表明，纳米波与衰减振荡解之间发生分岔的参数值取决于离散化的选择。最后，通过比较四阶Karpman方程的不同高阶离散化，我们表明分岔值对于大阶数趋于一个非零常数，而不是像相关的连续Karpman方程那样趋于$0$。
英文摘要：
We consider generalizations of nonlinear Schrödinger equations, which we call "Karpman equations", that include additional linear higher-order derivatives. Singularly-perturbed Karpman equations produce generalized solitary waves (GSWs) in the form of solitary waves with exponentially small oscillatory tails. Nanoptera are a special case of GSWs in which these oscillatory tails do not decay. Previous research on continuous third-order and fourth-order Karpman equations has shown that nanoptera occur in specific settings. We use exponential asymptotic techniques to identify traveling nanoptera in singularly-perturbed continuous Karpman equations. We then study the effect of discretization on nanoptera by applying a finite-difference discretization to continuous Karpman equations and studying traveling-wave solutions. The finite-difference discretization turns a continuous Karpman equation into an advance--delay equation, which we study using exponential asymptotic analysis. By comparing nanoptera in these discrete Karpman equations with nanoptera in their continuous counterparts, we show that the oscillation amplitudes and periods in the nanoptera tails differ in the continuous and discretized equations. We also show that the parameter values at which there is a bifurcation between nanopteron and decaying oscillatory solutions depends on the choice of discretization. Finally, by comparing different higher-order discretizations of the fourth-order Karpman equation, we show that the bifurcation value tends to a nonzero constant for large orders, rather than to $0$ as in the associated continuous Karpman equation.
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