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

TL;DR

The paper investigates nanoptera in higher-order Karpman equations, combining continuous and discrete models to understand exponentially small oscillations in traveling waves. Using exponential asymptotics, it derives singulant and prefactor structures that govern Stokes switching and the emergence of non-decaying tails, validating results with numerical simulations. Discretization is shown to shift bifurcation values and tail characteristics, with critical thresholds depending on parity and discretization order, though high-order discretizations recover the continuous behavior. The work highlights a discrete Peierls–Nabarro–type barrier in lattice Karpman systems and provides a framework to compare continuous and lattice wave dynamics across discretizations, with implications for optical and lattice systems modeled by higher-order NLS equations.

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. 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.

Figures (14)

• Figure 1: Solitary waves and generalized solitary waves (GSWs), which we denote by $w(x,t)$. (a) A solitary wave, which is spatially localized. (b) A symmetric nanopteron, which has non-vanishing, exponentially small oscillations in its tails. (c) A radiatively decaying GSW, which has spatially localized oscillations that are exponentially small.
• Figure 2: Stokes curves in the solution of the continuous fourth-order Karpman equation \ref{['KARP']} with the leading-order focusing solitary wave solution \ref{['LESNLS']}. The Stokes curves both follow the dashed line (so they overlap). Each Stokes curve arises from one singularity of the leading-order solution. These singularities are located at $x = V t \pm \mathrm{i}\pi/(2A)$, which we show as red circles. The Stokes multipliers $\mathcal{S}_1$ and $\mathcal{S}_2$ take different values on the two sides of the Stokes curves. We shade one of these regions to distinguish between the two regions.
• Figure 3: Stokes structure of a GSW solution $\psi$ of \ref{['KARP']}. The singularities are endpoints of the Stokes curves. The Stokes curves are lines that are parallel to the imaginary axis; they move with velocity $V$ along with the core of a GSW. The Stokes multipliers $\mathcal{S}_1$ and $\mathcal{S}_2$ change their values in a neighborhood of width $\mathcal{O}(\sqrt{\epsilon})$ around the Stokes curves.
• Figure 4: Simulated solution of the continuous fourth-order Karpman equation \ref{['KARP']} for $\epsilon = 0.1$ and an initial condition $\psi_0(x,0)$ that is given by \ref{['LESNLS']}. We show the solution at $t = 0$ and at four evenly-spaced points in time (which we round to two decimal places). The vertical scale is for the solution at $t = 159.32$. We vertically offset the solutions at other times to clearly convey the wave propagation. The solution has a leading-order solitary wave that propagates to the right and transient radiation that propagates to the left. Exponentially small oscillations, which we magnify in the inset, form behind the leading-order solitary wave.
• Figure 5: The averaged amplitude $|\psi_{\text{exp}}|_{\text{avg}}$ of the exponentially small oscillatory tails of the asymptotic and numerical GSWs for $\lambda =1$, $A=1$, and $V=1$ using logarithmic axes. The asymptotic and numerical results are consistent, and the asymptotic estimate becomes more accurate as $\epsilon \rightarrow 0$.