13 research outputs found
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