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J. Douglas Wright

Publications and source records attributed to J. Douglas Wright.

At least 19 recordsLinked to original sources
arXiv2025

Radiating Solitary Waves in an FPUT Lattice with Random Coefficients

We study the propagation of solitary waves in a Fermi-Pasta-Ulam-Tsingou (FPUT) lattice with small random heterogeneity in the linear spring force. Perturbed by the random environment, solitary waves lose energy through a radiative tail, resulting in gradual amplitude attenuation. As long as the wave remains coherent, we track its position and amplitude via a modulation approach. An expansion of the resulting modulation equations provides explicit predictions for the slow average amplitude decay, which we verify through numerical simulations.

math.DS↗
arXiv2024

Small-amplitude periodic traveling waves in dimer Fermi-Pasta-Ulam-Tsingou lattices

We prove the existence of small-amplitude periodic traveling waves in dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattices without assumptions of physical symmetry. Such lattices are infinite, one-dimensional chains of coupled particles in which the particle masses and/or the potentials of the coupling springs can alternate. Previously, periodic traveling waves were constructed in a variety of limiting regimes for the symmetric mass and spring dimers, in which only one kind of material data alternates. The new results discussed here remove the symmetry assumptions by exploiting the gradient structure and translation invariance of the traveling wave problem. Together, these features eliminate certain solvability conditions that symmetry would otherwise manage and facilitate a bifurcation argument involving a two-dimensional kernel.

math.DS↗
arXiv2024

Ion-Acoustic Wave Dynamics in a Two-Fluid Plasma

Plasma is a medium containing free electrons and cations, where each particle group behaves as a conducting fluid with a single velocity and temperature in the presence of electromagnetic fields. The difference in roles electrons and ions play define the two-fluid description of plasma. This paper examines ion-acoustic waves generated by the particles in both hot and cold plasma using a collisionless "Euler-Poisson" (EP) system. Employing phase-space asymptotic analysis, we establish that for specific wave speeds, EP acquires homoclinic orbits at the steady-state equilibrium and consequently, traveling waves. Combining python and Wolfram Mathematica, we captured visualizations of such behavior in one spatial dimension.

physics.plasm-ph↗
arXiv2024

Long-Wavelength Limit of the Two-Fluid Euler-Poisson System

Plasma is a medium filled with free electrons and positive ions. Each particle acts as a conducting fluid with a single velocity and temperature when electromagnetic fields are present. This distinction between the roles played by electrons and ions is what we refer to as the two$-$fluid description of plasma. In this paper, we investigate the dynamics of these particles in both hot and cold plasma using a collisionless ''Euler-Poisson'' system. Employing analytical and computational techniques from differential equations, we show this system is governed by the dynamics of the Korteweg$-$de Vries (KdV) equation in the long$-$wavelength limit.

physics.plasm-ph↗
arXiv2023

Approximation of (some) FPUT lattices by KdV Equations

We consider a Fermi-Pasta-Ulam-Tsingou lattice with randomly varying coefficients. We discover a relatively simple condition which when placed on the nature of the randomness allows us to prove that small amplitude/long wavelength solutions are almost surely rigorously approximated by solutions of Korteweg-de Vries equations for very long times. The key ideas combine energy estimates with homogenization theory and the technical proof requires a novel application of autoregressive processes.

math.AP↗
arXiv2022

A simple model of radiating solitary waves

To understand an oft-observed but poorly understood phenomenon in which a solitary wave in a dispersive equation slowly deteriorates due to a persistent emission of radiation (i.e. a ``radiating solitary wave''), we propose a bare-bones model which captures many essential features and which we are capable of analyzing completely by way of the Laplace transform. We find that wave amplitude decreases at an exponential rate but with a decay constant that is (in many cases) small beyond all orders of the frequency.

math.AP↗
arXiv2021

Using Random Walks to Establish Wavelike Behavior in an FPUT System with Random Coefficients

We consider a linear Fermi-Pasta-Ulam-Tsingou lattice with random spatially varying material coefficients. Using the methods of stochastic homogenization we show that solutions with long wave initial data converge in an appropriate sense to solutions of a wave equation. The convergence is strong and both almost sure and in expectation, but the rate is quite slow. The technique combines energy estimates with powerful classical results about random walks, specifically the law of the iterated logarithm.

math.AP↗
arXiv2020

Well-posedness and asymptotics of a coordinate-free model of flame fronts

We investigate a coordinate-free model of flame fronts introduced by Frankel and Sivashinsky; this model has a parameter $α$ which relates to how unstable the front might be. We first prove short-time well-posedness of the coordinate-free model, for any value of $α>0.$ We then argue that near the threshold $α\approx 1,$ the solution stays arbitrarily close to the solution of the weakly nonlinear Kuramoto--Sivashinsky (KS) equation, as long as the initial values are close.

math.AP↗
arXiv2020

Mass-in-Mass Lattices with Small Internal Resonators

We consider the mass-in-mass (MiM) lattice when the internal resonators are very small. When there are no internal resonators the lattice reduces to a standard Fermi-Pasta-Ulam-Tsingou (FPUT) system. We show that the solution of the MiM system, with suitable initial data, shadows the FPUT system for long periods of time. Using some classical oscillatory integral estimates we can conclude that the error of the approximation is (in some settings) higher than one may expect.

math.AP↗
arXiv2020

Solitary Waves in Mass-in-Mass Lattices

We consider the existence of spatially localized traveling wave solutions of the mass-in-mass lattice. Under an anti-resonance condition first discovered by Kevrekidis, Stefanov and Xu, we prove that such solutions exist in two distinguished limits, the first where the mass of the internal resonator is small and the second where the internal spring is very stiff. We then numerically simulate the solutions and these simulations indicate that the anti-resonant traveling waves are weakly unstable

math.AP↗
arXiv2018

Well-posedness of fully nonlinear KdV-type evolution equations

We study the well-posedness of the initial value problem for fully nonlinear evolution equations, $u_{t}=f[u],$ where $f$ may depend on up to the first three spatial derivatives of $u.$ We make three primary assumptions about the form of $f:$ a regularity assumption, a dispersivity assumption, and an assumption related to the strength of backwards diffusion. Because the third derivative of $u$ is present in the right-hand side and we effectively assume that the equation is dispersive, we say that these fully nonlinear evolution equations are of KdV-type. We prove the well-posedness of the initial value problem in the Sobolev space $H^{7}(\mathbb{R}).$ The proof relies on gauged energy estimates which follow after making two regularizations, a parabolic regularization and mollification of the initial data.

math.AP↗
arXiv2018

Generalized Solitary Waves in the Gravity-Capillary Whitham Equation

We study the existence of traveling wave solutions to a unidirectional shallow water model which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between $0$ and ${1}/{3}$) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave. We also present numerical evidence, based on the recent analytical work of Hur \& Johnson, that the asymptotic end states are modulationally stable for all Bond numbers between $0$ and $1/3$.

math.AP↗
arXiv2018

Existence theory for magma equations in dimension two and higher

We examine a degenerate, dispersive, nonlinear wave equation related to the evolution of partially molten rock in dimensions two and higher. This simplified model, for a scalar field capturing the melt fraction by volume, has been studied by direct numerical simulation where it has been observed to develop stable solitary waves. In this work, we prove local in time well-posedness results for the time dependent equation, on both the whole space and the torus, for dimensions two and higher. We also prove the existence of the solitary wave solutions in dimensions two and higher.

math.AP↗
arXiv2018

Small amplitude traveling waves in the full-dispersion Whitham equation

In this article, we provide an alternative way to construct small amplitude traveling waves for general Whitham type equations, in both periodic and whole line contexts. More specifically, Fourier analysis techniques allow us to reformulate the problem to the study of waves that are small and regular perturbations of well-understood ODE's. In addition, rigorous stability of these waves is established.

math.AP↗
arXiv2017

Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio

Consider an infinite chain of masses, each connected to its nearest neighbors by a (nonlinear) spring. This is a Fermi-Pasta-Ulam-Tsingou lattice. We prove the existence of traveling waves in the setting where the masses alternate in size. In particular we address the limit where the mass ratio tends to zero. The problem is inherently singular and we find that the traveling waves are not true solitary waves but rather "nanopterons", which is to say, waves which asymptotic at spatial infinity to very small amplitude periodic waves. Moreover, we can only find solutions when the mass ratio lies in a certain open set. The difficulties in the problem all revolve around understanding Jost solutions of a nonlocal Schrödinger operator in its semi-classical limit.

math.AP↗
arXiv2016

Stability of twisted states in the continuum Kuramoto model

We study a nonlocal diffusion equation approximating the dynamics of coupled phase oscillators on large graphs. Under appropriate assumptions, the model has a family of steady state solutions called twisted states. We prove a sufficient condition for stability of twisted states with respect to perturbations in the Sobolev and BV spaces. As an application, we study stability of twisted states in the Kuramoto model on small-world graphs.

nlin.PS↗