Elliot Blackstone
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Elliot Blackstone has a total of 15 co-authors. Here are the top co-authors by number of shared publications:
- Peter D. Miller
Department of Mathematics, University of Michigan, Ann Arbor, USA
4 shared publications Christophe Charlier
Department of Mathematics, KTH Royal Institute of Technology, 10044 Stockholm, Sweden
2 shared publications- Christophe Charlier
Lund University, Sweden
2 shared publications Jonatan Lenells
Department of Mathematics, KTH Royal Institute of Technology, 10044 Stockholm, Sweden
2 shared publications- Jonatan Lenells
Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden
2 shared publications - Matthew Dominique Mitchell
Department of Physics, University of Michigan, Ann Arbor, USA
2 shared publications
Latest Publications
- Top Cited PapersOpen Access
- 23 February 2026
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 407 (3) , 60
Universality in the Small-Dispersion Limit of the Benjamin–Ono Equation
We examine the solution of the Benjamin–Ono Cauchy problem for rational initial data in three types of double-scaling limits in which the dispersion tends to zero while simultaneously the independent variables either approach a point on one of the two branches of the caustic curve of the inviscid Burgers equation, or approach the critical point where the branches meet. The results reveal universal limiting profiles in each case that are independent of details of the initial data. We compare the results obtained with corresponding results for the Korteweg-de Vries equation found by Claeys–Grava in three papers (Claeys and Grava in Commun Math Phys 286:979–1009, 2009, Commun Pure Appl Math 63:203–232, 2010, SIAM J Math Anal 42:2132–2154, 2010). Our method is to analyze contour integrals appearing in an explicit representation of the solution of the Cauchy problem, in various limits involving coalescing saddle points.
- 23 February 2026
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 407 (3) , 60
Top Cited PapersOpen AccessUniversality in the Small-Dispersion Limit of the Benjamin–Ono Equation
We examine the solution of the Benjamin–Ono Cauchy problem for rational initial data in three types of double-scaling limits in which the dispersion tends to zero while simultaneously the independent variables either approach a point on one of the two branches of the caustic curve of the inviscid Burgers equation, or approach the critical point where the branches meet. The results reveal universal limiting profiles in each case that are independent of details of the initial data. We compare the results obtained with corresponding results for the Korteweg-de Vries equation found by Claeys–Grava in three papers (Claeys and Grava in Commun Math Phys 286:979–1009, 2009, Commun Pure Appl Math 63:203–232, 2010, SIAM J Math Anal 42:2132–2154, 2010). Our method is to analyze contour integrals appearing in an explicit representation of the solution of the Cauchy problem, in various limits involving coalescing saddle points.
- 24 October 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
The Benjamin–Ono initial-value problem for rational data with application to long-time asymptotics and scattering
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.
- 24 October 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
The Benjamin–Ono initial-value problem for rational data with application to long-time asymptotics and scattering
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.
- Open Access
Universality in the Small-Dispersion Limit of the Benjamin-Ono Equation
We examine the solution of the Benjamin-Ono Cauchy problem for rational initial data in three types of double-scaling limits in which the dispersion tends to zero while simultaneously the independent variables either approach a point on one of the two branches of the caustic curve of the inviscid Burgers equation, or approach the critical point where the branches meet. The results reveal universal limiting profiles in each case that are independent of details of the initial data. We compare the results obtained with corresponding results for the Korteweg-de Vries equation found by Claeys-Grava in three papers. Our method is to analyze contour integrals appearing in an explicit representation of the solution of the Cauchy problem, in various limits involving coalescing saddle points.
Open AccessUniversality in the Small-Dispersion Limit of the Benjamin-Ono Equation
We examine the solution of the Benjamin-Ono Cauchy problem for rational initial data in three types of double-scaling limits in which the dispersion tends to zero while simultaneously the independent variables either approach a point on one of the two branches of the caustic curve of the inviscid Burgers equation, or approach the critical point where the branches meet. The results reveal universal limiting profiles in each case that are independent of details of the initial data. We compare the results obtained with corresponding results for the Korteweg-de Vries equation found by Claeys-Grava in three papers. Our method is to analyze contour integrals appearing in an explicit representation of the solution of the Cauchy problem, in various limits involving coalescing saddle points.
- Open Access
The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves
The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin-Ono equation in $L^2(\mathbb{R})$ is obtained explicitly for generic rational initial data $u_0$. An explicit asymptotic wave profile $u^\mathrm{ZD}(t,x;ε)$ is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data $u_0$, such that the solution $u(t,x;ε)$ of the Benjamin-Ono equation with dispersion parameter $ε>0$ and initial data $u_0$ satisfies $u(t,x;ε)-u^\mathrm{ZD}(t,x;ε)\to 0$ in the locally uniform sense as $ε\to 0$, provided a discriminant inequality holds implying that certain caustic curves in the $(t,x)$-plane are avoided. In some cases this convergence implies strong $L^2(\mathbb{R})$ convergence. The asymptotic profile $u^\mathrm{ZD}(t,x;ε)$ is consistent with the modulated multi-phase wave solutions described by Dobrokhotov and Krichever.Open AccessThe Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves
The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin-Ono equation in $L^2(\mathbb{R})$ is obtained explicitly for generic rational initial data $u_0$. An explicit asymptotic wave profile $u^\mathrm{ZD}(t,x;ε)$ is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data $u_0$, such that the solution $u(t,x;ε)$ of the Benjamin-Ono equation with dispersion parameter $ε>0$ and initial data $u_0$ satisfies $u(t,x;ε)-u^\mathrm{ZD}(t,x;ε)\to 0$ in the locally uniform sense as $ε\to 0$, provided a discriminant inequality holds implying that certain caustic curves in the $(t,x)$-plane are avoided. In some cases this convergence implies strong $L^2(\mathbb{R})$ convergence. The asymptotic profile $u^\mathrm{ZD}(t,x;ε)$ is consistent with the modulated multi-phase wave solutions described by Dobrokhotov and Krichever. - Open Access
- 2 November 2023
- Published by Wiley in Mathematische Nachrichten
- Vol. 297 (4) , 1328-1369
On singular limits of finite Hilbert transform operators on multi‐intervals
- 2 November 2023
- Published by Wiley in Mathematische Nachrichten
- Vol. 297 (4) , 1328-1369
Open AccessOn singular limits of finite Hilbert transform operators on multi‐intervals
- Open Access
- 15 August 2023
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- Vol. 154 (5) , 1431-1472
Toeplitz determinants with a one-cut regular potential and Fisher–Hartwig singularities I. Equilibrium measure supported on the unit circle
We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential $V$ , (ii) Fisher–Hartwig singularities and (iii) a smooth function in the background. The potential $V$ is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials $V$ , the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher–Hartwig singularities. For non-constant $V$ , our results appear to be new even in the case of no Fisher–Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.- 15 August 2023
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- Vol. 154 (5) , 1431-1472
Open AccessToeplitz determinants with a one-cut regular potential and Fisher–Hartwig singularities I. Equilibrium measure supported on the unit circle
We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential $V$ , (ii) Fisher–Hartwig singularities and (iii) a smooth function in the background. The potential $V$ is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials $V$ , the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher–Hartwig singularities. For non-constant $V$ , our results appear to be new even in the case of no Fisher–Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble. - Open Access
Toeplitz determinants with a one-cut regular potential and Fisher--Hartwig singularities I. Equilibrium measure supported on the unit circle
We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential $V$, (ii) Fisher--Hartwig singularities, and (iii) a smooth function in the background. The potential $V$ is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials $V$, the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher--Hartwig singularities. For non-constant $V$, our results appear to be new even in the case of no Fisher--Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.Open AccessToeplitz determinants with a one-cut regular potential and Fisher--Hartwig singularities I. Equilibrium measure supported on the unit circle
We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential $V$, (ii) Fisher--Hartwig singularities, and (iii) a smooth function in the background. The potential $V$ is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials $V$, the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher--Hartwig singularities. For non-constant $V$, our results appear to be new even in the case of no Fisher--Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble. - Open Access
- 11 September 2021
- Published by World Scientific Pub Co Pte Ltd in Random Matrices: Theory and Applications
- Vol. 11 (02)
Gap probabilities in the bulk of the Airy process
- 11 September 2021
- Published by World Scientific Pub Co Pte Ltd in Random Matrices: Theory and Applications
- Vol. 11 (02)
Open AccessGap probabilities in the bulk of the Airy process
- 26 August 2020
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2022 (4) , 2636-2687
Oscillatory Asymptotics for the Airy Kernel Determinant on Two Intervals
We obtain asymptotics for the Airy kernel Fredholm determinant on two intervals. We give explicit formulas for all the terms up to and including the oscillations of order $1$, which are expressed in terms of Jacobi $\theta $-functions.- 26 August 2020
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2022 (4) , 2636-2687
Oscillatory Asymptotics for the Airy Kernel Determinant on Two Intervals
We obtain asymptotics for the Airy kernel Fredholm determinant on two intervals. We give explicit formulas for all the terms up to and including the oscillations of order $1$, which are expressed in terms of Jacobi $\theta $-functions.- Open Access
- 10 June 2020
- Published by Springer Nature in Analysis and Mathematical Physics
- Vol. 10 (3) , 1-59
Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach
In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $$\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])$$ and $$\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])$$ . These operators arise when one studies the interior problem of tomography. The diagonalization of $$\mathcal {H}_R,\mathcal {H}_L$$ has been previously obtained, but only asymptotically when $$b_L\ne -b_R$$ . We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes $$\mathcal {H}_R,\mathcal {H}_L$$ explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.- 10 June 2020
- Published by Springer Nature in Analysis and Mathematical Physics
- Vol. 10 (3) , 1-59
Open AccessDiagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach
In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $$\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])$$ and $$\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])$$ . These operators arise when one studies the interior problem of tomography. The diagonalization of $$\mathcal {H}_R,\mathcal {H}_L$$ has been previously obtained, but only asymptotically when $$b_L\ne -b_R$$ . We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes $$\mathcal {H}_R,\mathcal {H}_L$$ explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.