Abstract:
We present a model problem for benchmarking codes that investigate magma migration in the Earth's interior. This system retains the essential features of more sophisticated models, yet has the advantage of possessing solitary wave solutions. The existence of such exact solutions to the nonlinear problem make it an excellent benchmark problem for combinations of solver algorithms. In this work, we explore a novel algorithm for computing high quality approximations of the solitary waves and use them to benchmark a semi-Lagrangian Crank-Nicholson scheme for a finite element discretization of the time dependent problem.

Abstract:
The focusing cubic nonlinear Schr\"odinger equation in two dimensions admits vortex solitons, standing wave solutions with spatial structure, Qm(r,theta) = e^{i m theta} Rm(r). In the case of spin m = 1, we prove there exists a class of data that collapse with the vortex soliton profile at the log-log rate. This extends the work of Merle and Rapha\"el, (the case m = 0,) and suggests that the L2 mass that may be concentrated at a point during generic collapse may be unbounded. Difficulties with m >= 2 or when breaking the spin symmetry are discussed.

Abstract:
A common challenge to proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving stability. Following the work of Marzuola & Simpson, we prove the absence of embedded eigenvalues for a collection of nonlinear Schrodinger equations, including some one and three dimensional supercritical equations, and the three dimensional cubic-quintic equation. Our results also rule out nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances. The proof is computer assisted as it depends on the sign of certain inner products which do not readily admit analytic representations. Our source code is available for verification at http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.

Abstract:
We consider an equation for a thin-film of fluid on a rotating cylinder and present several new analytical and numerical results on steady state solutions. First, we provide an elementary proof that both weak and classical steady states must be strictly positive so long as the speed of rotation is nonzero. Next, we formulate an iterative spectral algorithm for computing these steady states. Finally, we explore a non-existence inequality for steady state solutions from the recent work of Chugunova, Pugh, & Taranets.

Abstract:
Parallel replica dynamics is a method for accelerating the computation of processes characterized by a sequence of infrequent events. In this work, the processes are governed by the overdamped Langevin equation. Such processes spend much of their time about the minima of the underlying potential, occasionally transitioning into different basins of attraction. The essential idea of parallel replica dynamics is that the exit time distribution from a given well for a single process can be approximated by the minimum of the exit time distributions of $N$ independent identical processes, each run for only 1/N-th the amount of time. While promising, this leads to a series of numerical analysis questions about the accuracy of the exit distributions. Building upon the recent work in Le Bris et al., we prove a unified error estimate on the exit distributions of the algorithm against an unaccelerated process. Furthermore, we study a dephasing mechanism, and prove that it will successfully complete.

Abstract:
One way of getting insight into non-Gaussian measures, posed on infinite dimensional Hilbert spaces, is to first obtain good approximations in terms of Gaussians. These best fit Gaussians then provide notions of mean and variance, and they can be used to accelerate sampling algorithms. This begs the question of how one should measure optimality. Here, we consider the problem of minimizing the distance between a family of Gaussians and the target measure, with respect to relative entropy, or Kullback-Leibler divergence, as has been done previously in the literature. Thus, it is desirable to have algorithms, well posed in the abstract Hilbert space setting, which converge to these minimizers. We examine this minimization problem by seeking roots of the first variation of relative entropy, taken with respect to the mean of the Gaussian, leaving the covariance fixed. We prove the convergence of Robbins-Monro type root finding algorithms, highlighting the assumptions necessary for them to converge to relative entropy minimizers.

Abstract:
Coherent structures, such as solitary waves, appear in many physical problems, including fluid mechanics, optics, quantum physics, and plasma physics. A less studied setting is found in geophysics, where highly viscous fluids couple to evolving material parameters to model partially molten rock, magma, in the Earth's interior. Solitary waves are also found here, but the equations lack useful mathematical structures such as an inverse scattering transform or even a variational formulation. A common question in all of these applications is whether or not these structures are stable to perturbation. We prove that the solitary waves in this Earth science setting are asymptotically stable and accomplish this without any pre-exisiting Lyapunov stability. This holds true for a family of equations, extending beyond the physical parameter space. Furthermore, this extends existing results on well-posedness to data in a neighborhood of the solitary waves.

Abstract:
We consider the one-dimensional propagation of electromagnetic waves in a weakly nonlinear and low-contrast spatially inhomogeneous medium with no energy dissipation. We focus on the case of a periodic medium, in which dispersion enters only through the (Floquet-Bloch) spectral band dispersion associated with the periodic structure; chromatic dispersion (time-nonlocality of the polarization) is neglected. Numerical simulations show that for initial conditions of wave-packet type (a plane wave of fixed carrier frequency multiplied by a slow varying, spatially localized function) very long-lived spatially localized coherent soliton-like structures emerge, whose character is that of a slowly varying envelope of a train of shocks. We call this structure an envelope carrier-shock train. The structure of the solution violates the oft-assumed nearly monochromatic wave packet structure, whose envelope is governed by the nonlinear coupled mode equations (NLCME). The inconsistency and inaccuracy of NLCME lies in the neglect of all (infinitely many) resonances except for the principle resonance induced by the initial carrier frequency. We derive, via a nonlinear geometrical optics expansion, a system of nonlocal integro-differential equations governing the coupled evolution of backward and forward propagating waves. These equations incorporate effects of all resonances. In a periodic medium, these equations may be expressed as a system of infinitely many coupled mode equations, which we call the extended nonlinear coupled mode system (xNLCME). Truncating xNLCME to include only the principle resonances leads to the classical NLCME. Numerical simulations of xNLCME demonstrate that it captures both large scale features, related to third harmonic generation, and fine scale carrier shocks features of the nonlinear periodic Maxwell equations.

Abstract:
In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schr\"odinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for the three dimensional cubic equation. Though we focus on a proof of the 3d cubic problem, this work presents a new algorithm for verifying certain spectral properties needed to study soliton stability. Source code for verification of our comptuations, and for further experimentation, are available at http://www.math.toronto.edu/simpson/files/spec_prop_code.tgz.

Abstract:
We study a derivative nonlinear Schr\"{o}dinger equation, allowing non-integer powers in the nonlinearity, $|u|^{2\sigma} u_x$. Making careful use of the energy method, we are able to establish short-time existence of solutions with initial data in the energy space, $H^1$. For more regular initial data, we establish not just existence of solutions, but also well-posedness of the initial value problem. These results hold for real-valued $\sigma\geq 1,$ while prior existence results in the literature require integer-valued $\sigma$ or $\sigma$ sufficiently large ($\sigma \geq 5/2$), or use higher-regularity function spaces.