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We consider generalizations of nonlinear Schr\"odinger equations, which wecall "Karpman equations", that include additional linear higher-orderderivatives. Singularly-perturbed Karpman equations produce generalizedsolitary waves (GSWs) in the form of solitary waves with exponentially smalloscillatory tails. Nanoptera are a special case of GSWs in which theseoscillatory tails do not decay. Previous research on continuous third-order andfourth-order Karpman equations has shown that nanoptera occur in specificsettings. We use exponential asymptotic techniques to identify travelingnanoptera in singularly-perturbed continuous Karpman equations. We then studythe effect of discretization on nanoptera by applying a finite-differencediscretization to continuous Karpman equations and studying traveling-wavesolutions. The finite-difference discretization turns a continuous Karpmanequation into an advance--delay equation, which we study using exponentialasymptotic analysis. By comparing nanoptera in these discrete Karpman equationswith nanoptera in their continuous counterparts, we show that the oscillationamplitudes and periods in the nanoptera tails differ in the continuous anddiscretized equations. We also show that the parameter values at which there isa bifurcation between nanopteron and decaying oscillatory solutions depends onthe choice of discretization. Finally, by comparing different higher-orderdiscretizations of the fourth-order Karpman equation, we show that thebifurcation value tends to a nonzero constant for large orders, rather than to$0$ as in the associated continuous Karpman equation.

Nanoptera In Higher-Order Nonlinear Schr\"odinger Equations: Effects Of Discretization