[Paper Review] Nanoptera and Stokes Curves in the 2-Periodic Fermi-Pasta-Ulam-Tsingou Equation

Christopher J. Lustri|arXiv (Cornell University)|May 17, 2019

Nonlinear Photonic Systems56 references15 citations

TL;DR

This paper uses exponential asymptotics to analyze nanoptera—nonlocal solitary waves with exponentially small oscillatory tails—in a 2-periodic Fermi-Pasta-Ulam-Tsingou lattice with small mass ratio. It shows that these oscillations arise via Stokes phenomenon when complex Stokes curves are crossed, and derives an asymptotic condition that predicts mass ratios where the oscillations cancel, yielding fully localized solitary waves. The results are validated numerically and extend exponential asymptotic methods to systems with approximate leading-order solutions.

ABSTRACT

This work presents asymptotic solutions to a singularly-perturbed, period-2 FPUT lattice and uses exponential asymptotics to examine `nanoptera', which are nonlocal solitary waves with constant-amplitude, exponentially small wave trains which appear behind the wave front. Using an exponential asymptotic approach, this work isolates the exponentially small oscillations, and demonstrates that they appear as special curves in the analytically-continued solution, known as `Stokes curves' are crossed. By isolating these the asymptotic form of these oscillations, it is shown that there are special mass ratios which cause the oscillations to vanish, producing localized solitary-wave solutions. The asymptotic predictions are validated through comparison with numerical simulations.

Motivation & Objective

• To understand the origin and structure of nanoptera—solitary waves with exponentially small oscillatory tails—in a diatomic FPUT lattice with small mass ratio.
• To investigate how exponentially small oscillations emerge in the far field via Stokes phenomenon in the complex plane.
• To derive an asymptotic condition that predicts when these oscillations vanish, resulting in fully localized solitary waves.
• To validate the asymptotic predictions through numerical simulations of the lattice dynamics.
• To extend exponential asymptotic techniques to systems where the leading-order solution is not exact but approximated via long-wave limits.

Proposed method

• Applies exponential asymptotics to a singularly perturbed, 2-periodic FPUT lattice with small mass ratio δ ≪ 1.
• Uses analytic continuation of the leading-order solution to identify singularities and Stokes curves in the complex plane.
• Derives asymptotic expressions for exponentially small oscillations using late-order terms and the method of dominant balance.
• Identifies the switching of oscillations across Stokes curves as the mechanism for their appearance in the solution.
• Compares asymptotic predictions of oscillation amplitude and nanopteron-free mass ratios with numerical simulations using a customized time-stepping scheme.
• Employs a Beale ansatz framework and long-wave approximation from prior work to model the leading-order behavior.

Experimental results

Research questions

• RQ1 How do exponentially small oscillations arise in the far field of supersonic traveling waves in a 2-periodic FPUT lattice?
• RQ2 What role does Stokes phenomenon play in the emergence of nanoptera in this system?
• RQ3 Can an asymptotic condition be derived that predicts mass ratios where the oscillatory tails vanish, yielding localized solitary waves?
• RQ4 How accurate are the asymptotic predictions of oscillation amplitude and nanopteron-free mass ratios compared to numerical simulations?
• RQ5 To what extent can exponential asymptotic methods be applied to systems with approximate leading-order solutions rather than exact ones?

Key findings

• The exponentially small oscillations in nanoptera arise due to Stokes phenomenon, appearing when Stokes curves are crossed in the complex plane.
• An asymptotic expression for the oscillation amplitude is derived, valid in the far field, given by equation (51).
• A critical condition for the cancellation of oscillations—equation (58)—is derived, predicting mass ratios where nanoptera become fully localized solitary waves.
• Numerical simulations confirm the asymptotic predictions: for K = 2 to 6, the computed δ values for nanopteron-free solutions closely match the asymptotic predictions, with relative errors below 10%.
• The condition for cancellation arises from destructive interference between two distinct exponentially small wave trains, each activated upon crossing a different Stokes curve.
• The method successfully predicts localized solutions in both one-sided (metastable) and two-sided (truly stable) nanopteron families, showing that the cancellation condition is independent of wave symmetry.