Asymptotic approximations to travelling waves in the diatomic Fermi-Pasta-Ulam lattice

We construct high-order approximate travelling waves solutions of the diatomic Fermi-Pasta-Ulam lattice using asymptotic techniques which are valid for arbitrary mass ratios. Separately small amplitude ansatzs are made for the motion of the lighter and heavier particles, which are coupled The Fredholm alternative is used to derive consistency conditions, whose solution generates small amplitude expansions for both sets of particles

Micropterons, Nanopterons and Solitary Wave Solutions to the Diatomic Fermi-Pasta-Ulam-Tsingou Problem

We use a specialized boundary-value problem solver for mixed-type functional differential equations to numerically examine the landscape of traveling wave solutions to the diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) problem. By using a continuation approach, we are able to uncover the relationship between the branches of micropterons and nanopterons that have been rigorously constructed recently in various limiting regimes. We show that the associated surfaces are connected together in a nontrivial fashion and illustrate the key role that solitary waves play in the branch points. Finally, we numerically show that the diatomic solitary waves are stable under the full dynamics of the FPUT system

Locating complex singularities of Burgers' equation using exponential asymptotics and transseries

Burgers' equation is an important mathematical model used to study gas dynamics and traffic flow, among many other applications. Previous analysis of solutions to Burgers' equation shows an infinite stream of simple poles born at t = 0^+, emerging rapidly from the singularities of the initial condition, that drive the evolution of the solution for t > 0. We build on this work by applying exponential asymptotics and transseries methodology to an ordinary differential equation that governs the small-time behaviour in order to derive asymptotic descriptions of these poles and associated zeros. Our analysis reveals that subdominant exponentials appear in the solution across Stokes curves; these exponentials become the same size as the leading order terms in the asymptotic expansion along anti-Stokes curves, which is where the poles and zeros are located. In this region of the complex plane, we write a transseries approximation consisting of nested series expansions. By reversing the summation order in a process known as transasymptotic summation, we study the solution as the exponentials grow, and approximate the pole and zero location to any required asymptotic accuracy. We present the asymptotic methods in a systematic fashion that should be applicable to other nonlinear differential equations.Comment: 30 pages, 6 figure

An energy-based stability criterion for solitary traveling waves in Hamiltonian lattices

In this work, we revisit a criterion, originally proposed in [Nonlinearity {\bf 17}, 207 (2004)], for the stability of solitary traveling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-traveling frame, as well as at that of the Floquet problem arising when considering the traveling wave as a periodic orbit modulo a shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the traveling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed

Micropterons, nanopterons and solitary wave solutions to the diatomic Fermi–Pasta–Ulam–Tsingou problem

Analysis and Stochastic