arXiv subjects

Peter Howard

Publications and source records attributed to Peter Howard.

15 recordsLinked to original sources

arXiv2026

A New Measure of Coarseness for Solutions to Cahn--Hilliard Equations

We introduce a new measure of coarseness for characterizing phase separation processes such as those described by Cahn--Hilliard equations. An advantage of our measure is that it remains consistent throughout the evolution, including for solutions with no periodic structure. We use our measure to compare two previous models of coarsening dynamics with numerically generated dynamics, providing the first direct check that we are aware of for the efficacy of these methods.

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arXiv2025

Renormalized oscillation theory for singular linear Hamiltonian pencils

For many applications, critical information about system dynamics is encoded in associated eigenvalue problems that can be posed as linear Hamiltonian systems with suitable boundary conditions. Motivated by examples from hydrodynamics, quantum mechanics, and magnetohydrodynamics (MHD), we develop a general framework for analyzing a broad class of linear Hamiltonian systems with at least one singular boundary condition and possible nonlinear dependence on the spectral parameter. We show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of $\mathbb{C}^{2n}$. This extends previous work by the authors for regular linear Hamiltonian systems that depend nonlinearly on the spectral parameter and singular linear Hamiltonian systems that depend linearly on the spectral parameter. We conclude the study by using our framework to study the spectrum in the setting of each of our motivating examples.

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arXiv2024

Comparison of Coarsening Dynamics for the Cahn--Hilliard and Burgers--Cahn--Hilliard Equations

We consider coarsening dynamics associated with a Burgers--Cahn--Hilliard system modeling a two-phase flow in one space dimension. Our emphasis is on the effect that coupling between the phase and fluid dynamics has on coarsening rates, and on the mechanisms driving this effect. We start with a detailed examination of coarsening dynamics for the uncoupled Cahn--Hilliard equation, comparing numerically generated rates with two analytic methods, and then we consider how these dynamics are affected by appropriate coupling with a viscous Burgers equation. In order to keep the analysis as self-contained as possible, we establish the global well-posedness of the system under consideration.

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arXiv2023

Oscillation Theory and Instability of Nonlinear Waves

In recent work, Baird et al. have introduced a generalized Maslov index which allows oscillation techniques that have previously been restricted to eigenvalue problems with underlying Hamiltonian structure to be extended to the non-Hamiltonian setting [T. J. Baird, P. Cornwell, G. Cox, C. Jones, and R. Marangell, Generalized Maslov indices for non-Hamiltonian systems, SIAM J. Math. Anal. 54 (2022) 1623-1668]. We show that this approach can be implemented in the analysis of spectral instability for nonlinear waves, taking as our setting a class of equations previously investigated by Pego and Weinstein via the Evans function [R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Phil. Trans. R. Soc. Lond. A 340 (1992) 47-94].

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arXiv2022

Renormalized oscillation theory for regular linear non-Hamiltonian systems

In recent work, Baird et al. have generalized the definition of the Maslov index to paths of Grassmannian subspaces that are not necessarily contained in the Lagrangian Grassmannian [T. J. Baird, P. Cornwell, G. Cox, C. Jones, and R. Marangell, {\it Generalized Maslov indices for non-Hamiltonian systems}, SIAM J. Math. Anal. {\bf 54} (2022) 1623-1668]. Such an extension opens up the possibility of applications to non-Hamiltonian systems of ODE, and Baird and his collaborators have taken advantage of this observation to establish oscillation-type results for obtaining lower bounds on eigenvalue counts in this generalized setting. In the current analysis, the author shows that renormalized oscillation theory, appropriately defined in this generalized setting, can be applied in a natural way, and that it has the advantage, as in the traditional setting of linear Hamiltonian systems, of ensuring monotonicity of crossing points as the independent variable increases for a wide range of system/boundary-condition combinations. This seems to mark the first effort to extend the renormalized oscillation approach to the non-Hamiltonian setting.

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arXiv2020

The Maslov index and spectral counts for linear Hamiltonian systems on $\mathbb{R}$

Working with a general class of linear Hamiltonian systems specified on $\mathbb{R}$, we develop a framework for relating the Maslov index to the number of eigenvalues the systems have on intervals of the form $[\lambda_1, \lambda_2)$ and $(-\infty, \lambda_2)$. We verify that our framework can be implemented for Sturm-Liouville systems, fourth-order potential systems, and a family of systems nonlinear in the spectral parameter. The analysis is primarily motivated by applications to the analysis of spectral stability for nonlinear waves, and aspects of such analyses are emphasized.

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arXiv2020

Renormalized Oscillation Theory for Singular Linear Hamiltonian Systems

Working with a general class of linear Hamiltonian systems with at least one singular boundary condition, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of $\mathbb{C}^{2n}$. This extends previous work by the authors for regular linear Hamiltonian systems.

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arXiv2019

The Maslov and Morse Indices for Sturm-Liouville Systems on the Half-Line

We show that for Sturm-Liouville Systems on the half-line $[0,\infty)$, the Morse index can be expressed in terms of the Maslov index and an additional term associated with the boundary conditions at $x = 0$. Relations are given both for the case in which the target Lagrangian subspace is associated with the space of $L^2 ((0,\infty), \mathbb{C}^{n})$ solutions to the Sturm-Liouville System, and the case when the target Lagrangian subspace is associated with the space of solutions satisfying the boundary conditions at $x = 0$. In the former case, a formula of H\"ormander's is used to show that the target space can be replaced with the Dirichlet space, along with additional explicit terms. We illustrate our theory by applying it to an eigenvalue problem that arises when the nonlinear Schr\"odinger equation on a star graph is linearized about a half-soliton solution.

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arXiv2018

Renormalized oscillation theory for linear Hamiltonian systems on [0,1] via the Maslov index

Working with a general class of linear Hamiltonian systems on $[0, 1]$, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of $\mathbb{C}^{2n}$. We verify that our applicability class includes Dirac and Sturm-Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly.

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arXiv2016

The Maslov and Morse indices for Schr\"odinger operators on $\mathbb{R}$

Assuming a symmetric potential that approaches constant endstates with a sufficient asymptotic rate, we relate the Maslov and Morse indices for Schr\"odinger operators on $\mathbb{R}$. In particular, we show that with our choice of convention, the Morse index is precisely the negative of the Maslov index.

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arXiv2016

The Maslov index for Lagrangian pairs on $\mathbb{R}^{2n}$

We discuss a definition of the Maslov index for Lagrangian pairs on $\mathbb{R}^{2n}$ based on spectral flow, and develop many of its salient properties. We provide two applications to illustrate how our approach leads to a straightforward analysis of the relationship between the Maslov index and the Morse index for Sch\"odinger operators on $[0,1]$ and $\mathbb{R}$.

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arXiv2016

The Maslov and Morse indices for Schrodinger operators on [0,1]

Assuming a symmetric potential and separated self-adjoint boundary conditions, we relate the Maslov and Morse indices for Schr\"odinger operators on $[0, 1]$. We find that the Morse index can be computed in terms of the Maslov index and two associated matrix eigenvalue problems. This provides an efficient way to compute the Morse index for such operators.

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arXiv2005

Sharp pointwise bounds for perturbed viscous shock waves

Refining previous work in \cite{Z.3, MaZ.3, Ra, HZ, HR}, we derive sharp pointwise bounds on behavior of perturbed viscous shock profiles for large-amplitude Lax or overcompressive type shocks and physical viscosity. These extend well-known results of Liu \cite{Liu97} obtained by somewhat different techniques for small-amplitude Lax type shocks and artificial viscosity, completing a program set out in \cite{ZH}. As pointed out in \cite{Liu91, Liu97}, the key to obtaining sharp bounds is to take account of cancellation associated with the property that the solution decays faster along characteristic than in other directions. Thus, we must here estimate characteristic derivatives for the entire nonlinear perturbation, rather than judicially chosen parts as in \cite{Ra, HR}. a requirement that greatly complicates the analysis.

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arXiv2005

Pointwise Asymptotic Behavior of Perturbed Viscous Shock Profiles

We consider the asymptotic behavior of perturbations of Lax and overcompressive type viscous shock profiles arising in systems of regularized conservation laws with strictly parabolic viscosity, and also in systems of conservation laws with partially parabolic regularizations such as arise in the case of the compressible Navier--Stokes equations and in the equations of magnetohydrodynamics. Under the necessary conditions of spectral and hyperbolic stability, together with transversality of the connecting profile, we establish detailed pointwise estimates on perturbations from a sum of the viscous shock profile under consideration and a family of diffusion waves which propagate perturbation signals along outgoing characteristics. Our approach combines the recent $L^p$-space analysis of Raoofi [$L^p$ Asympototic Behavior of Perturbed Viscous Shock Profiles, to appear J. Hyperbolic Differential Equations] with a straightforward bootstrapping argument that relies on a refined description of nonlinear signal interactions, which we develop through convolution estimates involving Green's functions for the linear evolutionary PDE that arises upon linearization of the regularized conservation law about the distinguished profile. Our estimates are similar to, though slightly weaker than, those developed by Liu in his landmark result on the case of weak Lax type profiles arising in the case of identity viscosity [Pointwise Convergence to Shock Waves for Viscous Conservation Laws, Comm. Pure Appl. Math. 50 (1997) 1113--1182].

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arXiv2004

Stability of undercompressive shock profiles

Using a simplified pointwise iteration scheme, we establish nonlinear phase-asymptotic orbital stability of large-amplitude Lax, undercompressive, overcompressive, and mixed under--overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations $|u_0(x)|\le E_0 (1+|x|)^{-3/2}$ in $C^{0+α}$, $E_0$ sufficiently small, under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile. This completes the program initiated by Zumbrun and Howard in \cite{ZH}, extending to the general undercompressive case results obtained for Lax and overcompressive shock profiles in \cite{SzX}, \cite{L}, \cite{ZH}, \cite{Z.2}, \cite{Ra}, \cite{MZ.1}--\cite{MZ.5}, and for special undercompressive profiles in \cite{LZ.1}--\cite{LZ.2}, \cite{HZ}. In particular, together with spectral results of \cite{Z.6}, our results yield nonlinear stability of large-amplitude undercompressive phase-transitional profiles near equilibrium of Slemrod's model \cite{Sl.5} for van der Waal gas dynamics or elasticity with viscosity--capillarity.

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