Exponential Asymptotics for Dark Solitons of the Discrete NLS Model

Abstract

In the present work we revisit the problem of the dark solitary wave pinned in the discrete nonlinear Schr{ö}dinger equation. In a number of recent studies, the methodology of exponential asymptotics was attempted to be utilized in this problem, however the results were not found to be fully in agreement with associated multiprecision numerical computations. Here we resolve this conundrum by finding precise exponential asymptotics for the pinned dark solitary waves. Moreover, we reconcile the relevant result with a general theory of pinned dark solitary waves in the {\it continuum} nonlinear Schr{ö}dinger equations in the presence of external potentials.

Figures (4)

• Figure 1: Stokes structure for the discrete solitary wave solution. The relevant Stokes line is shown as a solid blue line. There are additional singularities further along the imaginary axis in both directions, but the corresponding Stokes lines switch on asymptotic contributions that are exponentially small compared to those considered here.
• Figure 2: The eigenvalue of the intersite solution is shown both upon its multiplication by $h^{5/2} e^{\frac{\sqrt{2} \pi^2}{2 h}}$ (which renders its asymptotic form transparent) in the left panel and "as is" in the right panel. The numerical data are shown in blue, while the red dashed provides the exponential asymptotic prediction of Eq. \ref{['pred7']} and the dash-dotted black curve yields the leading order approximation of Eq. (\ref{['pred5']}).
• Figure 3: Similar results to Figure \ref{['fig:numerical-is2-is3']} but for the imaginary part of the complex eigenvalue associated with the onsite solution. The prediction is essentially the same as above (now on the imaginary axis) and the agreement of both the leading order asymptotics (\ref{['pred6']}) and of the next-order correction (\ref{['pred6-impr']}) documented again by the black, dash-dotted and the red dashed curves, respectively.
• Figure 4: Numerical results for the real (unstable) part of the complex eigenvalue of the onsite configuration. The leading-order asymptotics of Eq. (\ref{['pred6']}) is given by the black dash-dotted curve.