Exponential-asymptotic accuracy for internal-mode eigenvalue scaling in discrete lattices

Determine the dependence on the lattice spacing h, with exponential-asymptotic accuracy comparable to recent results for translational eigenvalues, of the internal linearization eigenmodes that bifurcate from the continuous spectrum (i.e., resonances at the spectral band edge that become point-spectrum eigenvalues) in discrete nonlinear dispersive lattices, specifically the discrete nonlinear Schrödinger equation and the discrete sine-Gordon equation.

Background

The paper resolves, via exponential asymptotics, the exponentially small eigenvalues associated with translational-invariance breaking for dark solitons in the discrete nonlinear Schrödinger equation, providing accurate prefactors and next-order corrections validated by multiprecision numerics.

Beyond translational modes, prior literature on discrete systems such as the discrete NLS and discrete sine-Gordon has documented internal modes that bifurcate from the continuous spectrum: resonances at the band edge that convert into point-spectrum eigenvalues. Earlier views suggested power-law bifurcations, but subsequent work established mixed exponential and algebraic behavior. However, the precise dependence of these internal-mode eigenvalues on lattice spacing h has not been characterized to the same level of accuracy achieved recently for translation modes using exponential asymptotics. The authors explicitly identify closing this accuracy gap as an open problem.