Peter D. Miller

Professor of Mathematics

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Publication List

Works Submitted or in Preparation:

• [S11] "The semiclassical modified nonlinear Schrödinger equation III: asymptotic analysis of the Cauchy problem. The hyperbolic region for transsonic initial data." [With J. DiFranco. In preparation.]
• [S10] "Avoided Hamiltonian-Hopf bifurcations." [With R. Kollár. In preparation.]
• [S9] "Semiclassical soliton ensembles for the three-wave resonant interaction equations: asymptotic behavior for small time." [With R. Buckingham and R. Jenkins. In preparation.]
• [S8] "Solutions of the focusing nonlinear Schrödinger equation interpolating between high-order solitons and rogue waves." [With D. Bilman. In preparation.]
• [S7] "The McCoy-Tracy-Wu solutions of the Painlevé-III equation revisited." [With H.-Y. Huang and C.-K. Law. In preparation.]
• [S6] "Long-time asymptotics for the Benjamin-Ono equation: new proofs for rational data." [With L. Gassot and P. Gérard. In preparation.]
• [S5] "Infinite-order solutions for the AKNS hierarchy and their asymptotic behavior under the mKdV flow." [With E. Blackstone, D. Bilman, and G. Young. In preparation.]
• [S4] "The Benjamin-Ono Equation in the Long-Time Limit: Linearized Self-Similar Universality." [With L. Gassot and P. Gérard. 33 pages. Submitted for publication in Pure and Applied Analysis.]
• [S3] "A proof of the soliton resolution conjecture for the Benjamin-Ono equation." [With L. Gassot and P. Gérard. 28 pages. Submitted for publication in the Annals of Mathematics.]
• [S2] "Suleimanov-Talanov self-focusing and the hierarchy of the focusing nonlinear Schrödinger equation." [With R. Buckingham and R. Jenkins. 100 pages. Submitted for publication in Arch. Rat. Mech. Anal.]
• [S1] "On stratification, large-scale tides, and temporal changes in surface tidal elevations: Two-layer analytical model." [With A. N. Wetzel, B. K. Arbic, I. Cerovecki, M. C. Hendershott, R. H. Karsten, and J. F. Molinari. 48 pages.]

Research Monographs:

• [M2] Discrete Orthogonal Polynomials: Asymptotics and Applications, Annals of Mathematics Studies series, volume 164, Princeton University Press, Princeton, 2007. [With J. Baik, T. Kriecherbauer, and K. T.-R. McLaughlin. 184 pages.]
• [M1] Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation, Annals of Mathematics Studies series, volume 154, Princeton University Press, Princeton, 2003. [With S. Kamvissis and K. T.-R. McLaughlin. 265 pages.]

Textbooks:

• [T2] "Perturbation theory and asymptotics," in The Princeton Companion to Applied Mathematics, N. Higham, M. Dennis, P. Glendinning, P. Martin, F. Santosa, and J. Tanner, eds., Princeton University Press, Princeton, NJ, 2015. [pages 208-218]
• [T1] Applied Asymptotic Analysis, Graduate Studies in Mathematics, Vol. 75, AMS Publications, Providence, 2006. [467 pages]

Book Reviews:

• [R2] Review of Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons by M. J. Ablowitz, SIAM Rev., 54, pages 837-841, 2012.
• [R1] Review of Asymptotics and Borel Summability by O. Costin, in J. Approx. Th., 162, pages 881-883, 2010.

Journal Publications:

• [J80] "Infinite-order rogue waves that are small (but not small in \(L^2\))." [With D. Bilman. 82 pages. To appear in SIAM Journal on Mathematical Analysis.]
• [J79] "The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves," Comm. Pure Appl. Math. (Online first, 2026).   DOI:  10.1002/cpa.70044.   [With E. Blackstone, L. Gassot, and P. Gérard. Open Access.]
• [J78] "Rigorous methods for Bohr-Sommerfeld quantization rules," EMS Surveys in Mathematical Sciences (Online first, 2026).   DOI:  10.4171/emss/114. [With J. Dong and G. Young.]
• [J77] "Universality in the small-dispersion limit of the Benjamin-Ono equation," Comm. Math. Phys. 407, article number 60, pages 1-69, 2026.   DOI:  10.1007/s00220-025-05506-z.   [With E. Blackstone and M. Mitchell. Open Access.]
• [J76] "The Benjamin-Ono initial-value problem for rational data with application to long-time asymptotics and scattering," Ann. Inst. H. Poincaré C Anal. Non Linéaire (Online first, 2025).   DOI:  10.4171/AIHPC/169.   [With E. Blackstone, L. Gassot, and P. Gérard.]
• [J75] "General rogue waves of infinite order: Exact properties, asymptotic behavior, and effective numerical computation," J. Nonlinear Waves 1, e19, pages 1-89, 2025.   DOI:  10.1017/jnw.2025.10018.   [With D. Bilman. Open Access.]
• [J74] "Zero-dispersion limit for the Benjamin-Ono equation," Séminaire Laurent-Schwartz - EDP et applications Centre de mathématiques Laurent Schwartz, 2023-2024, Exposé no. VI, 1-16, 2024. [With E. Blackstone, L. Gassot, and P. Gérard.]
• [J73] "On strong zero-dispersion asymptotics for Benjamin-Ono soliton ensembles," Contemporary Mathematics 823, pages 213-270, 2025.  DOI:  10.1090/conm/823/16498.  [With E. Blackstone and L. Gassot.]
• [J72] "Painlevé-III monodromy maps under the \(D_6\to D_8\) confluence and applications to the large-parameter asymptotics of rational solutions," SIGMA 20, 019 (77 pages), 2024.   DOI:  10.3842/SIGMA.2024.019.   [With A. Barhoumi, O. Lisovyy, and A. Prokhorov.]
• [J71] "Differential equations for approximate solutions of Painlevé equations: Application to the algebraic solutions of the Painlevé-III (\(D_7\)) equation," SIGMA 20, 008 (27 pages), 2024.   DOI:  10.3842/SIGMA.2024.008.   [With R. Buckingham.]
• [J70] "On the algebraic solutions of the Painlevé-III (\(D_7\)) equation," Physica D 441, 133493 (22 pages), 2022.  DOI:  10.1016/j.physd.2022.133493.  [With R. Buckingham.]          
• [J69] "Large-degree asymptotics of rational Painlevé-IV solutions by the isomonodromy method," Constr. Approx. 56, pages 233-443, 2022.   DOI: 10.1007/s00365-022-09586-1. [With R. Buckingham.]
• [J68] "Broader universality of rogue waves of infinite order," Physica D 435, 133289 (12 pages), 2022. DOI: 10.1016/j.physd.2022.133289. [With D. Bilman.]
• [J67] "Extreme superposition: High-order fundamental rogue waves in the far-field regime." Memoirs of the AMS 300, number 1505, pages (i)-(v) and 1-90, 2024.  [With D. Bilman.]
• [J66] "On the Maxwell-Bloch system in the sharp-line limit without solitons," Comm. Pure Appl. Math., 77, pages 457-542, 2024. DOI: 10.1002/cpa.22136. [With S. Li.]
• [J65] "Universality near the gradient catastrophe point in the semiclassical sine-Gordon equation," Comm. Pure Appl. Math. 75, pages 1517-1641, 2022. DOI: 10.1002/cpa.22018. [With B.-Y. Lu.]
• [J64] "Extreme superposition: rogue waves of infinite order and the Painlevé-III hierarchy." Duke Math. J. 169, number 4, pages 671-760, 2020. DOI: 10.1215/00127094-2019-0066. [With D. Bilman and L. Ling.]
• [J63] "Dispersive asymptotics for linear and integrable equations by the \(\overline{\partial}\) steepest descent method," in Nonlinear Dispersive Partial Differential Equations and Inverse Scattering (P. D. Miller, P. A. Perry, J.-C. Saut, and C. Sulem, eds.), Fields Institute Communications 83, pages 253-291, 2019. DOI: 10.1007/978-1-4939-9806-7_5. [With M. Dieng and K. D. T.-R. McLaughlin.]
• [J62] "Rational solutions of the Painlevé-III equation: large parameter asymptotics," Constr. Approx. 51, pages 123-224, 2020. DOI: 10.1007/s00365-019-09463-4. [With T. Bothner. Open Access.]
• [J61] "On the increasing tritronquée solutions of the Painlevé-II equation," SIGMA 14, 125 (38 pages), 2018.
• [J60] "A study of the direct spectral transform for the defocusing Davey-Stewartson-II equation in the semiclassical limit," Comm. Pure Appl. Math. 72, pages 1474-1547, 2019. DOI: 10.1002/cpa.21822. [With O. Assainova, C. Klein, and K. T.-R. McLaughlin.]
• [J59] "A robust inverse scattering transform for the focusing nonlinear Schrödinger equation," Comm. Pure Appl. Math. 72, pages 1722-1805, 2019. DOI: 10.1002/cpa.21819. [With D. Bilman.]
• [J58] "The diffusion equation with nonlocal data," J. Math. Anal. Appl. 466, pages 1119-1143, 2018. DOI: 10.1016/j.jmaa.2018.05.064. [With D. A. Smith.]
• [J57] "Rational solutions of the Painlevé-III equation," Stud. Appl. Math. 141, pages 626-679, 2018. DOI: 10.1111/sapm.12220. [With T. Bothner and Y. Sheng.]
• [J56] "Rational solutions of the Painlevé-II equation revisited," SIGMA 13, 065 (29 pages), 2017. [With Y. Sheng.]
• [J55] "Semiclassical soliton ensembles for the three-wave resonant interaction equations," Comm. Math. Phys. 354, pages 1015-1100, 2017. DOI: 10.1007/s00220-017-2897-7. [With R. Buckingham and R. Jenkins.]
• [J54] "Dispersive hydrodynamics: Preface," Physica D 333, pages 1-5, 2016. DOI: 10.1016/j.physd.2016.07.002. [With G. Biondini, G. A. El, and M. A. Hoefer.]
• [J53] "On the generation of dispersive shock waves," Physica D 333, pages 66-83, 2016. DOI: 10.1016/j.physd.2016.04.011.
• [J52] "On the spectral and modulational stability of periodic wavetrains for nonlinear Klein-Gordon equations," Bulletin of the Brazilian Mathematical Society 47, pages 417-429, 2016. ISSN 1678-7714. [With C. K. R. T. Jones, R. Marangell, and R. Plaza.]
• [J51] "The scattering transform for the Benjamin-Ono equation in the small-dispersion limit," Physica D 333, pages 185-199, 2016. DOI: 10.1016/j.physd.2015.07.012. [With A. N. Wetzel.]
• [J50] "Direct scattering for the Benjamin-Ono equation with rational initial data," Studies in Applied Mathematics 137, pages 53-69, 2016. DOI: 10.1111/sapm.12101. [With A. N. Wetzel.]
• [J49] "Dynamical Hamiltonian-Hopf instabilities of periodic traveling waves in Klein-Gordon equations," Physica D 308, pages 87-93, 2015. DOI: 10.1016/j.physd.2015.06.006. [With R. Marangell.]
• [J48] "Large-degree asymptotics of rational Painlevé-II functions. Critical behavior." Nonlinearity 28, pages 1539-1596, 2015. [With R. Buckingham.]
• [J47] "Initial-boundary value problems for the defocusing nonlinear Schrödinger equation in the semiclassical limit," Studies in Applied Mathematics 134, pages 276-362, 2015. DOI: 10.1111/sapm.12075. [With Z. Qin. Selected for "Highlights of the Year 2015"]
• [J46] "Spectral and modulational stability of periodic wavetrains for the nonlinear Klein-Gordon equation," Journal of Differential Equations 257, pages 4632-4703, 2014. [With C. K. R. T. Jones, R. Marangell, and R. Plaza.]
• [J45] "Large-degree asymptotics of rational Painlevé-II functions. Noncritical behavior," Nonlinearity 27, pages 2489-2577, 2014. [With R. Buckingham.]
• [J44] "Graphical Krein signature theory and Evans-Krein functions," SIAM Review 56, pages 73-123, 2014. [With R. Kollár. ]
• [J43] "On the stability analysis of periodic sine-Gordon traveling waves," Physica D 251, pages 63-74, 2013. [With C. K. R. T. Jones, R. Marangell, and R. Plaza.]
• [J42] "The semiclassical modified nonlinear Schrödinger equation II: asymptotic analysis of the Cauchy problem. The elliptic region for transsonic initial data," Contemporary Mathematics 593, pages 29-81, 2013. [With J. DiFranco.]
• [J41] "The sine-Gordon equation in the semiclassical limit: dynamics of fluxon condensates," Memoirs of the AMS 225, number 1059, pages (i)-(v) and 1-136, 2013. [With R. Buckingham.]
• [J40] "The sine-Gordon equation in the semiclassical limit: critical behavior near a separatrix," Journal d'Analyse Mathématique 118, pages 397-492, 2012. [With R. Buckingham.]
• [J39] "The Benjamin-Ono hierarchy with asymptotically reflectionless initial data in the zero-dispersion limit," Comm. Math. Sci. 10, pages 117-130, 2012. [With Z. Xu.]
• [J38] "On the modified nonlinear Schrödinger equation in the semiclassical limit: supersonic, subsonic, and transsonic behavior," Acta Math. Sci. 31, pages 2343-2377, 2011. [With J. DiFranco and B. Muite.]
• [J37] "On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data," Comm. Pure Appl. Math. 64, pages 205-270, 2011. [With Z. Xu.]
• [J36] "The ∂ steepest descent method for orthogonal polynomials on the real line with varying weights," Intern. Math. Res. Notices 2008, Article ID rnn075, pages 1-66, 2008. [With K. T.-R. McLaughlin.]
• [J35] "Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation," Physica D 237, pages 2296-2341, 2008. [With R. Buckingham.]
• [J34] "The semiclassical modified nonlinear Schrödinger equation I: modulation theory and spectral analysis," Physica D 237, pages 947-997, 2008. [With J. DiFranco.]
• [J33] "Locating the zeros of partial sums of e^z with Riemann-Hilbert methods," Contemporary Mathematics 458, pages 183-195, 2008. [With T. Kriecherbauer, A. B. J. Kuijlaars, and K. T.-R. McLaughlin.]
• [J32] "Riemann-Hilbert problems with lots of discrete spectrum," Contemporary Mathematics 458, pages 163-181, 2008.
• [J31] "The N-soliton of the focusing nonlinear Schrödinger equation for N large," Comm. Pure Appl. Math. 60, pages 951-1026, 2007. [With G. D. Lyng.]
• [J30] "The ∂ steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights," Intern. Math. Res. Papers 2006, Article ID 48673, pages 1-78, 2006. [With K. T.-R. McLaughlin.]
• [J29] "Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles: announcement of results," Intern. Math. Res. Not. 2003, No. 15, pages 821-858, 2003. [With J. Baik, T. Kriecherbauer, and K. D. T.-R. McLaughlin.]
• [J28] "Asymptotics of semiclassical soliton ensembles: rigorous justification of the WKB approximation," Intern. Math. Res. Notices 2002, No. 8, pages 383-454, 2002.
• [J27] "On the semiclassical limit for the focusing nonlinear Schrödinger equation: sensitivity to analytic properties of the initial data," R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458, pages 135-156, 2002. [With S. R. Clarke.]
• [J26] "Optimal tail estimates for directed last passage site percolation with geometric random variables," Advances Theor. Math. Phys. 5, pages 1207-1250, 2001. [With J. Baik, P. Deift, K. T.-R. McLaughlin, and X. Zhou.]
• [J25] "An exactly solvable model for the interaction of linear waves with Korteweg-de Vries solitons," SIAM J. Math. Anal. 33, pages 261-285, 2001. [With S. R. Clarke.]
• [J24] "Some remarks on a WKB method for the nonselfadjoint Zakharov-Shabat eigenvalue problem with analytic potentials and fast phase," Physica D 152/153, pages 145-162, 2001.
• [J23] "Linear guidance properties of solitonic Y-junction waveguides," Opt. Quantum Electron. 33, pages 19-54, 2001. [With J. A. Besley and N. N. Akhmediev.]
• [J22] "On the generation of solitons and breathers in the modified Korteweg-de Vries equation," CHAOS 10, pages 383-392, 2000. [With S. R. Clarke, R. Grimshaw, E. Pelinovsky, and T. Talipova.]
• [J21] "Soliton interactions in perturbed nonlinear Schrödinger equations," Phys. Rev. E 61, pages 7121-7133, 2000. [With J. A. Besley and N. N. Akhmediev.]
• [J20] "Metastability of breather modes of time-dependent potentials," Nonlinearity 13, pages 507-568, 2000. [With A. Soffer and M. I. Weinstein.]
• [J19] "A coupled Korteweg-de Vries system and mass exchanges among solitons," Physica Scripta 61, pages 518-525, 2000. [With P. L. Christiansen.]
• [J18] "On the semiclassical limit of the focusing nonlinear Schrödinger equation," Phys. Lett. A 247, pages 75-86, 1998. [With S. Kamvissis.]
• [J17] "Macroscopic dynamics in quadratic nonlinear lattices," Phys. Rev. E 57, pages 6038-6049, 1998. [With O. Bang.]
• [J16] "Self-similar evolution of self-written waveguides," Opt. Lett. 29, pages 268-270, 1998. [With T. M. Monro, L. Poladian, and C. M. de Sterke.]
• [J15] "Modal expansions and completeness relations for some time-dependent Schrödinger equations," Physica D 123, pages 513-524, 1998. [With N. N. Akhmediev.]
• [J14] "Periodic optical waveguides: exact floquet theory and spectral properties," Stud. Appl. Math. 101, pages 343-355, 1998. [With J. A. Besley and N. N. Akhmediev.]
• [J13] "Modes of periodic waveguides," Opt. Lett. 22, pages 1162-1164, 1997. [With J. A. Besley and N. N. Akhmediev.]
• [J12] "Effects of surface roughness on gratings," J. Opt. Soc. Am. B 14, pages 1339-1344, 1997. [With F. Ladouceur and L. Poladian.]
• [J11] "Vector nonlinear Schrödinger hierarchies as approximate Kadomtsev-Petviashvili hierarchies," Physica D 101, pages 17-26, 1997.
• [J10] "Optical conveyor belts: a new scheme for fiber communications," Opt. Lett. 21, pages 1132-1134, 1996. [With N. N. Akhmediev and A. Ankiewicz.]
• [J9] "Exploiting discreteness for switching in waveguide arrays," Opt. Lett. 21, pages 1105-1107, 1996. [With O. Bang.]
• [J8] "Modulation of multiphase waves in the presence of resonance," Physica D 92, pages 1-27, 1996. [With N. M. Ercolani and C. D. Levermore.]
• [J7] "Zero-crosstalk junctions made from dark solitons," Phys. Rev. E 53, pages 4137-4142, 1996.
• [J6] "Transfer matrices for multiport devices made from solitons," Phys. Rev. E 53, pages 4098-4106, 1996. [With N. N. Akhmediev.]
• [J5] "Do solitons exchange conserved quantities during collisions?," Phys. Rev. Lett. 76, pages 38-41, 1996. [With N. N. Akhmediev.]
• [J4] "Finite genus solutions to the Ablowitz-Ladik equations," Comm. Pure Appl. Math. 48, pages 1369-1440, 1995. [With N. M. Ercolani, I. M. Krichever, and C. D. Levermore.]
• [J3] "Macroscopic lattice dynamics," Physica D 79, pages 1-15, 1994. [With M. H. Hays and C. D. Levermore.]
• [J2] "The quantum discrete self-trapping equation in the Hartree approximation," Physica D 69, pages 18-32, 1993. [With E. Wright, J. C. Eilbeck, M. H. Hays, and A. C. Scott.]
• [J1] "Binding energies for discrete nonlinear Schrödinger equations," Physica Scripta 44, pages 509-516, 1991. [With A. C. Scott, J. Carr, and J. C. Eilbeck.]

Conference Proceedings:

• [C7] "Linear Y-junctions for modal splitting from periodic waveguides," 3rd Asia-Pacific Conference on Communications, Sydney, Australia, 1997. [With J. A. Besley and N. N. Akhmediev.]
• [C6] "X-junctions without cross-talk," Proceedings of the 21st Australian Conference on Optical Fibre Technology (ACOFT '96), Gold Coast, Queensland, Australia, 1996. [With N. N. Akhmediev.]
• [C5] "Power transfer in waveguide devices made from solitons," Nonlinear Guided Waves and their Applications, Cambridge, England. OSA Technical Digest Series, Volume 15, 1996. [With N. N. Akhmediev.]
• [C4] "Optical conveyor belts: a new scheme for fiber communications," Proceedings of the 20th Australian Conference on Optical Fibre Technology (ACOFT '95), Coolum Beach, Queensland, Australia, 1995. [With N. N. Akhmediev and A. Ankiewicz.]
• [C3] "Multi-port switching in waveguide arrays," Proceedings of the 20th Australian Conference on Optical Fibre Technology (ACOFT '95), Coolum Beach, Queensland, Australia, 1995. [With O. Bang.]
• [C2] "Exploiting discreteness for switching in arrays of nonlinear waveguides," Copenhagen Conference on Complex Dynamics in Spatially Extended Systems, Niels Bohr Institute, Copenhagen, Denmark, 1995. Physica Scripta T67, pages 26-29, 1996. [With O. Bang.]
• [C1] "Macroscopic behavior in the Ablowitz-Ladik equations," in Nonlinear Evolution Equations and Dynamical Systems NEEDS 94, Makhankov, et. al, eds, World Scientific, 1995.

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