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Jonatan Lenells

Publications and source records attributed to Jonatan Lenells.

At least 55 records · Page 3Linked to original sources
arXiv2020

The family of confluent Virasoro fusion kernels and a non-polynomial $q$-Askey scheme

We study the recently introduced family of confluent Virasoro fusion kernels $\mathcal{C}_k(b,\boldsymbolθ,σ_s,ν)$. We study their eigenfunction properties and show that they can be viewed as non-polynomial generalizations of both the continuous dual $q$-Hahn and the big $q$-Jacobi polynomials. More precisely, we prove that: (i) $\mathcal{C}_k$ is a joint eigenfunction of four different difference operators for any positive integer $k$, (ii) $\mathcal{C}_k$ degenerates to the continuous dual $q$-Hahn polynomials when $ν$ is suitably discretized, and (iii) $\mathcal{C}_k$ degenerates to the big $q$-Jacobi polynomials when $σ_s$ is suitably discretized. These observations lead us to propose the existence of a non-polynomial generalization of the $q$-Askey scheme. The top member of this non-polynomial scheme is the Virasoro fusion kernel (or, equivalently, Ruijsenaars' hypergeometric function), and its first confluence is given by the $\mathcal{C}_k$.

math-ph↗
arXiv2020

Non-chiral Intermediate Long Wave equation and inter-edge effects in narrow quantum Hall systems

We present a non-chiral version of the Intermediate Long Wave (ILW) equation that can model nonlinear waves propagating on two opposite edges of a quantum Hall system, taking into account inter-edge interactions. We obtain exact soliton solutions governed by the hyperbolic Calogero-Moser-Sutherland (CMS) model, and we give a Lax pair, a Hirota form, and conservation laws for this new equation. We also present a periodic non-chiral ILW equation, together with its soliton solutions governed by the elliptic CMS model.

math-ph↗
arXiv2020

The multiplicative constant for the Meijer-$G$ kernel determinant

We compute the multiplicative constant in the large gap asymptotics of the Meijer-G point process. This point process generalizes the Bessel point process and appears at the hard edge of Cauchy--Laguerre multi-matrix models and of certain product random matrix ensembles.

math-ph↗
arXiv2020

Confluent Conformal Blocks of the Second Kind

We construct confluent conformal blocks of the second kind of the Virasoro algebra. We also construct the Stokes transformations which map such blocks in one Stokes sector to another. In the BPZ limit, we verify explicitly that the constructed blocks and the associated Stokes transformations reduce to solutions of the confluent BPZ equation and its Stokes matrices, respectively. Both the confluent conformal blocks and the Stokes transformations are constructed by taking suitable confluent limits of the crossing transformations of the four-point Virasoro conformal blocks.

hep-th↗
arXiv2020

A periodic hexagon tiling model and non-Hermitian orthogonal polynomials

We study a one-parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling. The description of the asymptotic behavior can be separated into two regimes: the low and the high temperature regime. Our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there. For low temperatures, the disorded region consists of two disjoint ellipses. In the high temperature regime the two ellipses merge into a single simply connected region. At the transition from the low to the high temperature a tacnode appears. The key to our asymptotic study is a recent approach introduced by Duits and Kuijlaars providing a double integral representation for the correlation kernel. One of the factors in the integrand is the Christoffel-Darboux kernel associated to polynomials that satisfy non-Hermitian orthogonality relations with respect to a complex-valued weight on a contour in the complex plane. We compute the asymptotic behavior of these orthogonal polynomials and the Christoffel-Darboux kernel by means of a Riemann-Hilbert analysis. After substituting the resulting asymptotic formulas into the double integral we prove our main results by classical steepest descent arguments.

math-ph↗
arXiv2019

Asymptotic analysis of Dotsenko-Fateev integrals

We develop a method for evaluating asymptotics of certain contour integrals that appear in Conformal Field Theory under the name of Dotsenko-Fateev integrals and which are natural generalizations of the classical hypergeometric functions. We illustrate the method by establishing a number of estimates that are useful in the context of martingale observables for multiple Schramm-Loewner evolution processes.

math-ph↗
arXiv2019

Higher order large gap asymptotics at the hard edge for Muttalib--Borodin ensembles

We consider the limiting process that arises at the hard edge of Muttalib--Borodin ensembles. This point process depends on $θ> 0$ and has a kernel built out of Wright's generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form \begin{equation*} \mathbb{P}(\mbox{gap on } [0,s]) = C \exp \left( -a s^{2ρ} + b s^ρ + c \ln s \right) (1 + o(1)) \qquad \mbox{as }s \to + \infty, \end{equation*} where the constants $ρ$, $a$, and $b$ have been derived explicitly via a differential identity in $s$ and the analysis of a Riemann--Hilbert problem. Their method can be used to evaluate $c$ (with more efforts), but does not allow for the evaluation of $C$. In this work, we obtain expressions for the constants $c$ and $C$ by employing a differential identity in $θ$. When $θ$ is rational, we find that $C$ can be expressed in terms of Barnes' $G$-function. We also show that the asymptotic formula can be extended to all orders in $s$.

math-ph↗
arXiv2019

On the spectral problem associated with the time-periodic nonlinear Schrödinger equation

According to its Lax pair formulation, the nonlinear Schrödinger (NLS) equation can be expressed as the compatibility condition of two linear ordinary differential equations with an analytic dependence on a complex parameter. The first of these equations---often referred to as the \emph{$x$-part} of the Lax pair---can be rewritten as an eigenvalue problem for a Zakharov-Shabat operator. The spectral analysis of this operator is crucial for the solution of the initial value problem for the NLS equation via inverse scattering techniques. For space-periodic solutions, this leads to the existence of a Birkhoff normal form, which beautifully exhibits the structure of NLS as an infinite-dimensional completely integrable system. In this paper, we take the crucial steps towards developing an analogous picture for time-periodic solutions by performing a spectral analysis of the \emph{$t$-part} of the Lax pair with a periodic potential.

math.AP↗
arXiv2019

The KdV equation on the half-line: Time-periodicity and mass transport

The work presented here emanates from questions arising from experimental observations of the propagation of surface water waves. The experiments in question featured a periodically moving wavemaker located at one end of a flume that generated unidirectional waves of relatively small amplitude and long wavelength when compared with the undisturbed depth. It was observed that the wave profile at any point down the channel very quickly became periodic in time with the same period as that of the wavemaker. One of the questions dealt with here is whether or not such a property holds for model equations for such waves. In the present discussion, this is examined in the context of the Korteweg-de Vries equation using the recently developed version of the inverse scattering theory for boundary value problems put forward by Fokas and his collaborators. It turns out that the Korteweg-de Vries equation does possess the properly that solutions at a fixed point down the channel have the property of asymptotic periodicity in time when forced periodically at the boundary. However, a more subtle issue to do with conservation of mass fails to hold at the second order in a small parameter which is the typical wave amplitude divided by the undisturbed depth.

nlin.SI↗
arXiv2019

Schramm's formula and the Green's function for multiple SLE

We construct martingale observables for systems of multiple SLE curves by applying screening techniques within the CFT framework recently developed by Kang and Makarov, extended to admit multiple SLEs. We illustrate this approach by rigorously establishing explicit expressions for the Green's function and Schramm's formula in the case of two curves growing towards infinity. In the special case when the curves are `fused' and start at the same point, some of the formulas we prove were predicted in the physics literature.

math.PR↗
arXiv2019

Airy and Painlevé asymptotics for the mKdV equation

We consider the higher order asymptotics for the mKdV equation in the Painlevé sector. We first show that the solution admits a uniform expansion to all orders in powers of $t^{-1/3}$ with coefficients that are smooth functions of $x(3t)^{-1/3}$. We then consider the special case when the reflection coefficient vanishes at the origin. In this case, the leading coefficient which satisfies the Painlevé II equation vanishes. We show that the leading asymptotics is instead described by the derivative of the Airy function. We are also able to express the subleading term explicitly in terms of the Airy function.

math.AP↗
arXiv2019

Asymptotics for the Sasa--Satsuma equation in terms of a modified Painlevé II transcendent

We consider the initial-value problem for the Sasa-Satsuma equation on the line with decaying initial data. Using a Riemann-Hilbert formulation and steepest descent arguments, we compute the long-time asymptotics of the solution in the sector $|x| \leq M t^{1/3}$, $M$ constant. It turns out that the asymptotics can be expressed in terms of the solution of a modified Painlevé II equation. Whereas the standard Painlevé II equation is related to a $2 \times 2$ matrix Riemann-Hilbert problem, this modified Painlevé II equation is related to a $3 \times 3$ matrix Riemann--Hilbert problem.

math.AP↗
arXiv2018

Exact solution of a Neumann boundary value problem for the stationary axisymmetric Einstein equations

For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a {\it Dirichlet} boundary condition imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an explicit expression for the spacetime metric corresponding to the Bardeen-Wagoner uniformly rotating disk of dust. In this paper, we consider a similar boundary value problem for a rotating disk in which a {\it Neumann} boundary condition is imposed along the disk instead of a Dirichlet condition. Using the integrable structure of the Ernst equation, we are able to reduce the problem to a Riemann-Hilbert problem on a genus one Riemann surface. By solving this Riemann-Hilbert problem in terms of theta functions, we obtain an explicit expression for the Ernst potential. Finally, a Riemann surface degeneration argument leads to an expression for the associated spacetime metric.

math-ph↗
arXiv2018

The hyperbolic Ernst equation in a triangular domain

The collision of two plane gravitational waves in Einstein's theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann--Hilbert problem. The formulation of the Riemann--Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform. Our results are valid also for boundary data whose derivatives are unbounded at the triangle's corners---this level of generality is crucial for the application to colliding gravitational waves. Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann--Hilbert formalism can be explicitly inverted at the boundary. In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary.

math.AP↗
arXiv2018

Construction of solutions and asymptotics for the sine-Gordon equation in the quarter plane

We consider the sine-Gordon equation in laboratory coordinates in the quarter plane. The first part of the paper considers the construction of solutions via Riemann-Hilbert techniques. In addition to constructing solutions starting from given initial and boundary values, we also construct solutions starting from an independent set of spectral (scattering) data. The second part of the paper establishes asymptotic formulas for the quarter-plane solution $u(x,t)$ as $(x,t) \to \infty$. Assuming that $u(x,0)$ and $u(0,t)$ approach integer multiples of $2π$ as $x \to \infty$ and $t \to \infty$, respectively, we show that the asymptotic behavior is described by four asymptotic sectors. In the first sector (characterized by $x/t \geq 1$), the solution approaches a multiple of $2π$ as $x \to \infty$. In the third sector (characterized by $0 \leq x/t \leq 1$ and $t|x-t| \to \infty$), the solution asymptotes to a train of solitons superimposed on a radiation background. The second sector (characterized by $0 \leq x/t \leq 1$ and $x/t \to 1$) is a transition region and the fourth sector (characterized by $x/t \to 0$) is a boundary region. We derive precise asymptotic formulas in all sectors. In particular, we describe the interaction between the asymptotic solitons and the radiation background, and derive a formula for the solution's topological charge.

math.AP↗
arXiv2017

Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

The solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann-Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.

math.AP↗