Nanoptera in a Period-2 Toda Chain

TL;DR

This paper investigates nonlocal solitary waves called nanoptera in a period-2 Toda lattice using exponential asymptotics, revealing conditions for their existence and how they can vanish to produce localized solutions.

Contribution

It introduces a novel exponential asymptotic analysis of nanoptera in a period-2 Toda chain and derives anti-resonance conditions for wave-free solutions.

Findings

Derived a simple asymptotic expression for nanoptera.

Identified conditions under which nanoptera vanish, leading to localized solitary waves.

Established an anti-resonance condition in the small mass ratio limit.

Abstract

We study asymptotic solutions to a singularly-perturbed, period-2 Toda lattice and use exponential asymptotics to examine `nanoptera', which are nonlocal solitary waves with constant-amplitude, exponentially small wave trains. With this approach, we isolate the exponentially small, constant-amplitude waves, and we elucidate the dynamics of these waves in terms of the Stokes phenomenon. We find a simple asymptotic expression for the waves, and we study configurations in which these waves vanish, producing localized solitary-wave solutions. In the limit of small mass ratio, we derive a simple anti-resonance condition for the manifestation these wave-free solutions.