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Jonatan Lenells

Publications and source records attributed to Jonatan Lenells.

At least 37 records · Page 2Linked to original sources
arXiv2022

Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation

We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic $A$-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a Bäcklund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin-Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.

math-ph↗
arXiv2022

Disk counting statistics near hard edges of random normal matrices: the multi-component regime

We consider a two-dimensional point process whose points are separated into two disjoint components by a hard wall, and study the multivariate moment generating function of the corresponding disk counting statistics. We investigate the ``hard edge regime" where all disk boundaries are a distance of order $\frac{1}{n}$ away from the hard wall, where $n$ is the number of points. We prove that as $n \to + \infty$, the asymptotics of the moment generating function are of the form \begin{align*} & \exp \bigg(C_{1}n + C_{2}\ln n + C_{3} + \mathcal{F}_{n} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(n^{-\frac{3}{5}})\bigg), \end{align*} and we determine the constants $C_{1},\dots,C_{4}$ explicitly. The oscillatory term $\mathcal{F}_{n}$ is of order $1$ and is given in terms of the Jacobi theta function. Our theorems allow us to derive various precise results on the disk counting function. For example, we prove that the asymptotic fluctuations of the number of points in one component are of order $1$ and are given by an oscillatory discrete Gaussian. Furthermore, the variance of this random variable enjoys asymptotics described by the Weierstrass $\wp$-function.

math-ph↗
arXiv2022

Wilson loops in the abelian lattice Higgs model

We consider the lattice Higgs model on $\mathbb{Z}^4$, with structure group given by $ \mathbb{Z}_n $ for $ n \geq 2 $. We compute the expected value of the Wilson loop observable to leading order when the gauge coupling constant and hopping parameter are both sufficiently large. The leading order term is expressed in terms of a quantity arising from the related but much simpler $ \mathbb{Z}_n $ model, which reduces to the Ising model when $n=2$. As part of the proof, we construct a coupling between the lattice Higgs model and the $ \mathbb{Z}_n $ model.

math.PR↗
arXiv2022

Exponential moments for disk counting statistics of random normal matrices in the critical regime

We obtain large $n$ asymptotics for the $m$-point moment generating function of the disk counting statistics of the Mittag-Leffler ensemble. We focus on the critical regime where all disk boundaries are merging at speed $n^{-\smash{\frac{1}{2}}}$, either in the bulk or at the edge. As corollaries, we obtain two central limit theorems and precise large $n$ asymptotics of all joint cumulants (such as the covariance) of the disk counting function. Our results can also be seen as large $n$ asymptotics for $n\times n$ determinants with merging planar discontinuities.

math-ph↗
arXiv2022

The focusing NLS equation with step-like oscillating background: asymptotics in a transition zone

In a recent paper, we presented scenarios of long-time asymptotics for a solution of the focusing nonlinear Schrödinger equation whose initial data approach two different plane waves $A_j\mathrm{e}^{\mathrm{i}ϕ_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ at minus and plus infinity. In the shock case $B_1<B_2$ some scenarios include sectors of genus $3$, that is sectors $ξ_1<ξ<ξ_2$, $ξ:=\frac{x}{t}$ where the leading term of the asymptotics is expressed in terms of hyperelliptic functions attached to a Riemann surface $M(ξ)$ of genus $3$. The long-time asymptotic analysis in such a sector is performed in another recent paper. The present paper deals with the asymptotic analysis in a transition zone between two genus $3$ sectors $ξ_1<ξ<ξ_0$ and $ξ_0<ξ<ξ_2$. The leading term is expressed in terms of elliptic functions attached to a Riemann surface $\tilde{M}$ of genus $1$. A central step in the derivation is the construction of a local parametrix in a neighborhood of two merging branch points. We construct this parametrix by solving a model problem which is similar to the Riemann-Hilbert problem associated with the Painlevé IV equation.

math.AP↗
arXiv2022

The focusing NLS equation with step-like oscillating background: the genus 3 sector

We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching different plane waves $A_j\mathrm{e}^{\mathrm{i}ϕ_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as $x\to\pm\infty$. The goal is to determine the long-time asymptotics of the solution, according to the value of $ξ=x/t$. The general situation is analyzed in [7] where we develop the Riemann-Hilbert approach and detect different scenarios of asymptotic analysis, depending on the relationships between the parameters $A_1$, $A_2$, $B_1$, and $B_2$. In particular, in the shock case $B_1<B_2$, some scenarios include genus $3$ sectors, i.e., ranges of values of $ξ$ where the leading term of the asymptotics is given in terms of hyperelliptic functions attached to a Riemann surface $M(ξ)$ of genus three. The present paper is devoted to the complete asymptotic analysis in such a sector.

math.AP↗
arXiv2022

Spin generalizations of the Benjamin-Ono equation

We present new soliton equations related to the $A$-type spin Calogero-Moser (CM) systems introduced by Gibbons and Hermsen. These equations are spin generalizations of the Benjamin-Ono (BO) equation and the recently introduced non-chiral intermediate long-wave (ncILW) equation. We obtain multi-soliton solutions of these spin generalizations of the BO equation and the ncILW equation via a spin-pole ansatz where the spin-pole dynamics is governed by the spin CM system in the rational and hyperbolic cases, respectively. We also propose physics applications of the new equations, and we introduce a spin generalization of the standard intermediate long-wave equation which interpolates between the matrix Korteweg-de Vries equation, the Heisenberg ferromagnet equation, and the spin BO equation.

math-ph↗
arXiv2021

Wilson loops in finite Abelian lattice gauge theories

We consider lattice gauge theories on $\mathbb{Z}^4$ with Wilson action and structure group $\mathbb{Z}_n$. We compute the expectation of Wilson loop observables to leading order in the weak coupling regime, extending and refining a recent result of Chatterjee. Our proofs use neither duality relations nor cluster expansion techniques.

math.PR↗
arXiv2021

The Robin problem for the nonlinear Schrödinger equation on the half-line

We consider the nonlinear Schrödinger equation on the half-line $x \geq 0$ with a Robin boundary condition at $x = 0$ and with initial data in the weighted Sobolev space $H^{1,1}(\mathbb{R}_+)$. We prove that there exists a global weak solution of this initial-boundary value problem and provide a representation for the solution in terms of the solution of a Riemann--Hilbert problem. Using this representation, we obtain asymptotic formulas for the long-time behavior of the solution. In particular, by restricting our asymptotic result to solutions whose initial data are close to the initial profile of the stationary one-soliton, we obtain results on the asymptotic stability of the stationary one-solitons under any small perturbation in $H^{1,1}(\mathbb{R}_+)$. In the focusing case, such a result was already established by Deift and Park using different methods, and our work provides an alternative approach to obtain such results.

math.AP↗
arXiv2021

The "good" Boussinesq equation: a Riemann-Hilbert approach

We develop an inverse scattering transform formalism for the "good" Boussinesq equation on the line. Assuming that the solution exists, we show that it can be expressed in terms of the solution of a $3 \times 3$ matrix Riemann-Hilbert problem. The Riemann-Hilbert problem is formulated in terms of two reflection coefficients whose definitions involve only the initial data, and it has a form which makes it suitable for the evaluation of long-time asymptotics via Deift-Zhou steepest descent arguments.

math.AP↗
arXiv2021

Non-polynomial $q$-Askey scheme: integral representations, eigenfunction properties, and polynomial limits

We construct a non-polynomial generalization of the $q$-Askey scheme. Whereas the elements of the $q$-Askey scheme are given by $q$-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars' hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev's quantum dilogarithm, Woronowicz's quantum exponential, or Kurokawa's double sine function. We present the basic properties of all the elements of the scheme, including their integral representations, joint eigenfunction properties, and polynomial limits.

math.CA↗
arXiv2021

The hard-to-soft edge transition: exponential moments, central limit theorems and rigidity

The local eigenvalue statistics of large random matrices near a hard edge transitioning into a soft edge are described by the Bessel process associated with a large parameter $α$. For this point process, we obtain 1) exponential moment asymptotics, up to and including the constant term, 2) asymptotics for the expectation and variance of the counting function, 3) several central limit theorems and 4) a global rigidity upper bound.

math.PR↗
arXiv2021

On the non-chiral intermediate long wave equation II: periodic case

We study integrability properties of the non-chiral intermediate long wave (ncILW) equation with periodic boundary conditions. The ncILW equation was recently introduced by the authors as a parity-invariant relative of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) use the Hirota method to construct the periodic multi-soliton solutions, (d) derive a Bäcklund transformation, (e) use the Bäcklund transformation to obtain an infinite number of conservation laws.

nlin.SI↗
arXiv2021

On the non-chiral intermediate long wave equation

We study integrability properties of the non-chiral intermediate long wave equation recently introduced by the authors as a parity-invariant variant of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) derive a Bäcklund transformation, (d) use, separately, the Bäcklund transformation and the Lax representation to obtain an infinite number of conservation laws.

nlin.SI↗
arXiv2021

The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

The Bessel process models the local eigenvalue statistics near $0$ of certain large positive definite matrices. In this work, we consider the probability \begin{align*} \mathbb{P}\Big( \mbox{there are no points in the Bessel process on } (0,x_{1})\cup(x_{2},x_{3})\cup\cdots\cup(x_{2g},x_{2g+1}) \Big), \end{align*} where $0<x_{1}<\cdots<x_{2g+1}$ and $g \geq 0$ is any non-negative integer. We obtain asymptotics for this probability as the size of the intervals becomes large, up to and including the oscillations of order $1$. In these asymptotics, the most intricate term is a one-dimensional integral along a linear flow on a $g$-dimensional torus, whose integrand involves ratios of Riemann $θ$-functions associated to a genus $g$ Riemann surface. We simplify this integral in two generic cases: (a) If the flow is ergodic, we compute the leading term in the asymptotics of this integral explicitly using Birkhoff's ergodic theorem. (b) If the linear flow has certain "good Diophantine properties", we obtain improved estimates on the error term in the asymptotics of this integral. In the case when the flow is both ergodic and has "good Diophantine properties" (which is always the case for $g=1$, and "almost always" the case for $g \geq 2$), these results can be combined, yielding particularly precise and explicit large gap asymptotics.

math.PR↗
arXiv2020

Gap probabilities in the bulk of the Airy process

We consider the probability that no points lie on $g$ large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order $1$, and we prove this conjecture for $g=1$.

math.PR↗
arXiv2020

The focusing NLS equation with step-like oscillating background: scenarios of long-time asymptotics

We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching two different plane waves $A_j\mathrm{e}^{\mathrm{i}ϕ_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as $x\to\pm\infty$. Using Riemann-Hilbert techniques and Deift-Zhou steepest descent arguments, we study the long-time asymptotics of the solution. We detect that each of the cases $B_1 B_2$, and $B_1=B_2$ deserves a separate analysis. Focusing mainly on the first case, the so-called shock case, we show that there is a wide range of possible asymptotic scenarios. We also propose a method for rigorously establishing the existence of certain higher-genus asymptotic sectors.

math.AP↗