Nanoptera In Higher-Order Nonlinear Schrödinger Equations: Effects Of Discretization

We consider generalizations of nonlinear Schr\"odinger 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 osc...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Authors: Moston-Duggan, Aaron J, Porter, Mason A, Lustri, Christopher J
Format: Article
Language:English
Subjects:
Asymptotic properties
Bifurcations
Discretization
Finite difference method
Mathematical analysis
Schrodinger equation
Solitary waves
Traveling waves
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Summary:We consider generalizations of nonlinear Schr\"odinger 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 work on third-order and fourth-order Karpman equations has shown that nanoptera occur in specific continuous settings. We use exponential asymptotic techniques to identify traveling nanoptera in singularly-perturbed Karpman equations. We then study the effect of discretization on nanoptera by applying a finite-difference discretization to Karpman equations and using exponential asymptotic analysis to study traveling-wave solutions. By comparing nanoptera in lattice equations with nanoptera in their continuous counterparts, we show that the discretization process changes the amplitude and periodicity of the oscillations in nanoptera tails. We also show that discretization changes the parameter values at which there is a bifurcation between nanopteron and decaying oscillatory solutions. Finally, by comparing different higher-order discretizations of the fourth-order Karpman equation, we show that the bifurcation value tends to a nonzero constant as the order increases, rather than to \(0\) as in the associated continuous Karpman equation.
ISSN:2331-8422