Darryl H. Yong

SOLITARY WAVES IN LAYERED NONLINEAR MEDIA

We study longitudinal elastic strain waves in a one-dimensional periodically layered medium, alternating between two materials with different densities and stress-strain relations. If the impedances are different, dispersive effects are seen due to reflection at the interfaces. When the stress-strain relations are nonlinear, the combination of dispersion and nonlinearity leads to the appearance of solitary waves that interact like solitons. We study the scaling properties of these solitary waves and derive a homogenized system of equations that includes dispersive terms. We show that pseudospectral solutions to these equations agree well with direct solutions of the hyperbolic conservation laws in the layered medium using a high-resolution finite volume method. For particular parameters we also show how the layered medium can be related to the Toda lattice, which has discrete soliton solutions.

Nonlinear Dynamics of Mode-locking Optical Fiber Ring Lasers

We consider a model of a mode-locked fiber ring laser for which the evolution of a propagating pulse in a birefringent optical fiber is periodically perturbed by rotation of the polarization state owing to the presence of a passive polarizer. The stable modes of operation of this laser that correspond to pulse trains with uniform amplitudes are fully classified. Four parameters, i.e., polarization, phase, amplitude, and chirp, are essential for an understanding of the resultant pulse-train uniformity. A reduced set of four coupled nonlinear differential equations that describe the leading-order pulse dynamics is found by use of the variational nature of the governing equations. Pulse-train uniformity is achieved in three parameter regimes in which the amplitude and the chirp decouple from the polarization and the phase. Alignment of the polarizer either near the slow or the fast axis of the fiber is sufficient to establish this stable mode locking.

Strings, Chains, and Ropes

Following Antman [Amer. Math. Mon., 87 (1980), pp. 359-370], we advocate a more physically realistic and systematic derivation of the wave equation suitable for a typical undergraduate course in partial differential equations. To demonstrate the utility of this derivation, three applications that follow naturally are described: strings, hanging chains, and jump ropes.

Why No Difference? A Controlled Flipped Classroom Study for an Introductory Differential Equations Course

Abstract Flipped classrooms have the potential to improve student learning and metacognitive skills as a result of increased time for active learning and group work and student control over pacing, when compared with traditional lecture-based courses. We are currently running a 4-year controlled study to examine the impact of flipping an Introductory Differential Equations course at Harvey Mudd College. In particular, we compare flipped instruction with an interactive lecture with elements of active learning rather than a traditional lecture. The first two years of this study showed no differences in learning, metacognitive, or affective gains between the control and flipped sections. We believe that contextual factors, such as a strong group-work culture at Harvey Mudd College, contribute to the similar performance of both sections. Additionally, to maintain a rigorous experimental design, we maintained identical content across the control and flipped section; relaxing this requirement in a non-study setting would allow us to take further advantage of educational opportunities afforded by flipping, and may therefore improve student learning.

Phase Plane Behavior of Solitary Waves in Nonlinear Layered Media

The one-dimensional elastic wave equations for compressional waves have the form

Almost periodic factorization of certain block triangular matrix functions

\begin{array}{*{20}{c}} \hfill {{{ \in }_{t}}(x,t) - {{u}_{x}}(x,t) = 0} \\ \hfill {{{{(\rho (x)u(x,t))}}_{t}} - \sigma {{{( \in (x,t),x)}}_{x}} = 0} \\ \end{array}

Pulse-train uniformity and dynamics in optical fiber lasers

(1) where e(x, t) is the strain and u(x, t) the velocity. We consider a heterogeneous material with the density specified by ρ(x) and a nonlinear constitutive relation for the stress given by a function σ(∈, x) that also varies explicitly with x. This is a hyperbolic system of conservation laws with a spatially-varying flux function, q t + f(q, x) x = 0.

Nonlinear dynamics and pulse train uniformity of modelocking optical fiber ring lasers

We study how boundary conditions affect the multiple-scale analysis of hyperbolic conservation laws with rapid spatial fluctuations. The most significant difficulty occurs when one has insufficient boundary conditions to solve consistency conditions. We show how to overcome this missing boundary condition difficulty for both linear and nonlinear problems through the recovery of boundary information. We introduce two methods for this recovery (multiple-scale analysis with a reduced set of scales, and a combination of Laplace transforms and multiple scales) and show that they are roughly equivalent. We also show that the recovered boundary information is likely to contain secular terms if the initial conditions are nonzero. However, for the linear problem, we demonstrate how to avoid these secular terms to construct a solution that is valid for all time. For nonlinear problems, we argue that physically relevant problems do not exhibit the missing boundary condition difficulty.