About

Peter D. Miller is a professor at the University of Michigan Ann Arbor in the Department of Applied and Interdisciplinary Mathematics. He held a position as a professor at the University of Michigan from 1989 to 1994. His academic background includes a PhD from the University of Michigan, where his dissertation focused on Macroscopic Lattice Dynamics under the advisement of C. David Levermore. His research interests and contributions are centered around applied mathematics, with a focus on macroscopic lattice dynamics.

Research topics

• Mathematics
• Mathematical analysis
• Physics
• Mathematical physics
• Medicine

Selected publications

• On strong zero-dispersion asymptotics for Benjamin-Ono soliton ensembles

  Contemporary mathematics - American Mathematical Society · 2025-01-01 · 2 citations

  otherSenior author

  A soliton ensemble is a particular kind of approximation of the solution of an initial-value problem for an integrable equation by a reflectionless potential that is well adapted to singular asymptotics like the small-dispersion limit. We study soliton ensembles for the Benjamin-Ono equation by using reasonable hypotheses to develop local approximations that capture highly oscillatory features of the solution and hence provide more information than weak convergence results that are easier to obtain. These local approximations are deduced independently from empirically-observed but unproven distributions of eigenvalues of two related matrices, one Hermitian and another non-Hermitian. We perform careful numerical experiments to study the asymptotic behavior of the eigenvalues of these matrices in the small-dispersion limit, and formulate conjectures reflecting our observations. Then we apply the conjectures to construct the local approximations of slowly varying profiles and rapidly oscillating profiles as well. We show that the latter profiles are consistent with the predictions of Whitham modulation theory as originally developed for the Benjamin-Ono equation by Dobrokhotov and Krichever.

  PublisherDOI

• Suleimanov-Talanov self-focusing and the hierarchy of the focusing nonlinear Schrödinger equation

  ArXiv.org · 2025-07-02

  preprintOpen accessSenior author

  We study the self-focusing of wave packets from the point of view of the semiclassical focusing nonlinear Schrödinger equation. A type of finite-time collapse/blowup of the solution of the associated dispersionless limit was investigated by Talanov in the 1960s, and recently Suleimanov identified a special solution of the dispersive problem that formally regularizes the blowup and is related to the hierarchy of the Painlevé-III equation. In this paper we approximate the Talanov solutions in the full dispersive equation using a semiclassical soliton ensemble, a sequence of exact reflectionless solutions for a corresponding sequence of values of the semiclassical parameter epsilon tending to zero, approximating the Talanov initial data more and more accurately in the limit as epsilon tends to zero. In this setting, we rigorously establish the validity of the dispersive saturation of the Talanov blowup obtained by Suleimanov. We extend the result to the full hierarchy of higher focusing nonlinear Schrödinger equations, exhibiting new generalizations of the Talanov initial data that produce such dispersively regularized extreme focusing in both mixed and pure flows. We also argue that generic perturbations of the Talanov initial data lead to a different singularity of the dispersionless limit, namely a gradient catastrophe for which the dispersive regularization is instead based on the tritronquée solution of the Painlevé-I equation and the Peregrine breather solution which appears near points in space time corresponding to the poles of the former transcendental function as shown by Bertola and Tovbis.

  PublisherOA PDFDOI

• General rogue waves of infinite order: exact properties, asymptotic behaviour, and effective numerical computation

  Journal of Nonlinear Waves · 2025-01-01

  articleOpen accessSenior author

  Abstract This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schrödinger equations. We establish the following key property of these solutions: they are all in $L^2(\mathbb{R})$ with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this paper, we define general rogue waves of infinite order, establish their basic exact and asymptotic properties, and provide computational tools for calculating them accurately.

  PublisherDOI

• Infinite-order rogue waves that are small (but not small in $L^2$)

  ArXiv.org · 2025-07-31

  preprintOpen accessSenior author

  General rogue waves of infinite order constitute a family of solutions of the focusing nonlinear Schrödinger equation that have recently been identified in a variety of asymptotic limits such as high-order iteration of Bäcklund transformations and semiclassical focusing of pulses with specific amplitude profiles. These solutions have compelling properties such as finite $L^2$-norm contrasted with anomalously slow temporal decay in the absence of coherent structures. In this paper we investigate the asymptotic behavior of general rogue waves of infinite order in a parametric limit in which the solution becomes small uniformly on compact sets while the $L^2$-norm remains fixed. We show that the solution is primarily concentrated on one side of a specific curve in logarithmically rescaled space-time coordinates, and we obtain the leading-order asymptotic behavior of the solution in this region in terms of elliptic functions as well as near the boundary curve in terms of modulated solitons. The asymptotic formula captures the fixed $L^2$-norm even as the solution becomes uniformly small.

  PublisherOA PDFDOI

• Optical performance of Roman optical telescope assembly

  2025-09-17

  article

  The Roman Space Telescope is a three mirror anastigmat design with a 2.4 m primary mirror, which will provide science information on dark energy and exoplanets via two science instruments. The Optical Telescope Assembly (OTA) was aligned at ambient temperature in a vacuum chamber. Following a vibration test, the OTA’s optical performance, wavefront error, and pupil alignment, were tested at cold operational temperatures in a thermal vacuum chamber. Both test campaigns were performed in double-pass configuration, employing interferometry, Focus Diverse Phase Retrieval, and Shack-Hartman Wavefront Sensing test techniques. Alignment adjustments during the tests were made to the actuated Secondary Mirror, and Fold Mirror 1 in the Wide Field Instrument channel. Stability of the optical telescope assembly (OTA) is a critical need for Roman. For the TVAC test campaign, L3Harris and NASA developed a novel test with the objective of measuring the optical response of key parts of the OTA in the presence of small amplitude thermal stimuli; the Sinusoidal Thermoelastic Distortion Test (STDT). The basic concept is to use the flight hardware’s own thermal control system to generate repeated thermal disturbances while continuously monitoring the optical response. The telescope’s predicted On-Orbit, End of Life quasi static optical wave front error performance is well within the required 67.3/74.2 nm RMS for the WFI/CGI channel.

  PublisherDOI

• Rigorous Methods for Bohr-Sommerfeld Quantization Rules

  ArXiv.org · 2025-05-08

  preprintOpen access

  In this work, we prove Bohr-Sommerfeld quantization rules for the self-adjoint Zakharov-Shabat system and the Schrödinger equation in the presence of two simple turning points bounding a classically allowed region. In particular, we use the method of comparison equations for $2\times 2$ traceless first-order systems to provide a unified perspective that yields similar proofs in each setting. The use of a Weber model system gives results that are uniform in the eigenvalue parameter over the whole range from the bottom of the potential well up to finite values.

  PublisherOA PDFDOI

• The Benjamin–Ono initial-value problem for rational data with application to long-time asymptotics and scattering

  Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2025-10-24 · 2 citations

  articleOpen accessSenior author

  We show that the initial-value problem for the Benjamin–Ono equation on \mathbb{R} with L^{2}(\mathbb{R}) rational initial data with only simple poles can be solved in closed form via a determinant formula involving contour integrals. The dimension of the determinant depends on the number of simple poles of the rational initial data only and the matrix elements depend explicitly on the independent variables (t,x) and the dispersion coefficient \epsilon . This allows for various interesting asymptotic limits to be resolved quite efficiently. As an example, and as a first step towards establishing the soliton resolution conjecture, we prove that the solution with initial datum equal to minus a soliton exhibits scattering.

  PublisherDOI

• Preface to the special issue in memory of Hermann Flaschka

  Physica D Nonlinear Phenomena · 2024-03-27

  article
  PublisherDOI

• Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation

  Symmetry Integrability and Geometry Methods and Applications · 2024-01-20

  articleOpen accessSenior authorCorresponding

  It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$_7$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation.

  PublisherOA PDFDOI

• Extreme Superposition: High-Order Fundamental Rogue Waves in the Far-Field Regime

  Memoirs of the American Mathematical Society · 2024-08-07 · 14 citations

  articleSenior author

  This Memoir concerns fundamental rogue-wave solutions of the focusing nonlinear Schrödinger equation in the limit that the order of the rogue wave is large and the independent variables <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x comma t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x,t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are proportional to the order (the far-field limit). We first formulate a Riemann-Hilbert representation of these solutions that allows the order to vary continuously rather than by integer increments. The intermediate solutions in this continuous family include also soliton solutions for zero boundary conditions spectrally encoded by a single complex-conjugate pair of poles of arbitrary order, as well as other solutions having nonzero boundary conditions matching those of the rogue waves albeit with far slower decay as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow plus-or-minus normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mo> ± </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x\to \pm \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The large-order far-field asymptotic behavior of the solution depends on which of three disjoint regions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the “channels”), <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the “shelves”), and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper E"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the “exterior domain”) contains the rescaled variables. On the region <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the amplitude is small and the solution is highly oscillatory, while on the region <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the solution is approximated by a modulated plane wave with a highly oscillatory correction term. The asymptotic behavior on these two domains is the same for all continuous orders. Assuming that the order belongs to the discrete sequence characteristic of rogue-wave solutions, the asymptotic behavior of the solution on the region <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper E"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> resembles that on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>but without the oscillatory correction term</italic> . Solutions of other continuous orders behave quite differently on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper E"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

  PublisherDOI

Recent grants

• Applied Analysis for Integrable Nonlinear Waves

  NSF · $380k · 2015–2018

• Universality in Nonlinear Waves and Related Topics

  NSF · $350k · 2022–2026

• Frontiers in Asymptotic Analysis for Integrable Nonlinear Waves

  NSF · $315k · 2012–2016

• Asymptotic Problems in Nonlinear Waves and Beyond

  NSF · $300k · 2008–2012

• A Study of Wave Patterns

  NSF · $390k · 2018–2023

Frequent coauthors

• K. T-R McLaughlin

  19 shared

• Robert Buckingham

  University of Cincinnati

  18 shared

• C. David Levermore

  17 shared

• A. Leland Albright

  Indiana University – Purdue University Indianapolis

  15 shared

• Nicholas M. Ercolani

  13 shared

• I. M. Krichever

  Columbia University

  12 shared

• Betty Chow

  California Animal Hospital

  10 shared

• Robert Marangell

  10 shared

Education

• Ph.D. Applied Mathematics, Mathematics

  University of Arizona

  1994