Abstract

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. Previous research on continuous Karpman equations has shown that GSWs occur in specific settings. We use exponential asymptotic techniques to identify GSWs in singularly-perturbed continuous Karpman equations. We then study the effect of discretization on GSWs by applying a finite-difference discretization to continuous Karpman equations. By comparing GSWs in these discrete Karpman equations with GSWs in their continuous counterparts, we show that the oscillation amplitudes and periods in the GSWs differ in the continuous and discrete equations. We also show that the parameter values at which there is a bifurcation between GSW solutions and solitary wave 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. Co-Authors: Mason Porter and Christopher Lustri