arXiv subjects
Publications and source records attributed to Jonatan Lenells.
We consider the ``good" Boussinesq equation on the half-line. Assuming existence of the solution, we prove that it can be recovered from the solution of a $3\times 3$ Riemann-Hilbert problem that depends only on the initial and boundary values, and whose jump contour consists of twelve half-lines.
We propose an analytical method for the construction of Hartree-Fock phase diagrams for the (fermion) Hubbard model and various generalizations thereof. Such phase diagrams are traditionally constructed numerically, but we argue that, by using asymptotic techniques, it is possible to obtain analytic formulas approximating the curves separating the different phases to very high accuracy. To illustrate the new method, we apply it to the two-dimensional Hubbard model on the square lattice at zero temperature. This yields formulas for the Hartree-Fock phase boundaries that agree with, but also improve on, earlier numerical results. In particular, our results provide the first rigorous proof of the existence of mixed phases in this model.
We consider the antiferromagnetic gap for the half-filled two-dimensional (2D) Hubbard model (on a square lattice) at zero temperature in Hartree-Fock theory. It was conjectured by Hirsch in 1985 that this gap, $\Delta$, vanishes like $\exp(-2\pi\sqrt{t/U})$ in the weak-coupling limit $U/t\downarrow 0$ ($U>0$ and $t>0$ are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.
Let $\mu$ be a positive measure supported on a domain $\Omega$. We consider the behavior of the balayage measure $\nu:=\mathrm{Bal}(\mu,\partial \Omega)$ near a point $z_{0}\in \partial \Omega$ at which $\Omega$ has an outward-pointing cusp. Assuming that the order and coefficient of tangency of the cusp are $d>0$ and $a>0$, respectively, and that $d\mu(z) \asymp |z-z_{0}|^{2b-2}d^{2}z$ as $z\to z_0$ for some $b > 0$, we obtain the leading order term of $\nu$ near $z_{0}$. This leading term is universal in the sense that it only depends on $d$, $a$, and $b$. We also treat the case when the domain has multiple corners and cusps at the same point. Finally, we obtain an explicit expression for the balayage of the uniform measure on the tacnodal region between two osculating circles, and we give an application of this result to two-dimensional Coulomb gases.
We prove a semiclassical asymptotic formula for the two elements $\mathcal M$ and $\mathcal Q$ lying at the bottom of the recently constructed non-polynomial hyperbolic $q$-Askey scheme. We also prove that the corresponding exponent is a generating function of the canonical transformation between pairs of Darboux coordinates on the monodromy manifold of the Painlev\'e I and $\textrm{III}_3$ equations, respectively. Such pairs of coordinates characterize the asymptotics of the tau function of the corresponding Painlev\'e equation. We conjecture that the other members of the non-polynomial hyperbolic $q$-Askey scheme yield generating functions associated to the other Painlev\'e equations in the semiclassical limit.
We present a numerical scheme for the solution of the initial-value problem for the ``bad'' Boussinesq equation. The accuracy of the scheme is tested by comparison with exact soliton solutions as well as with recently obtained asymptotic formulas for the solution.
We consider the balayage of a measure $\mu$ defined on a domain $\Omega$ onto its boundary $\partial \Omega$. Assuming that $\Omega$ has a corner of opening $\pi \alpha$ at a point $z_0 \in \partial \Omega$ for some $0 < \alpha \leq 2$ and that $d\mu(z) \asymp |z-z_{0}|^{2b-2}d^{2}z$ as $z\to z_0$ for some $b > 0$, we obtain the precise rate of vanishing of the balayage of $\mu$ near $z_{0}$. The rate of vanishing is universal in the sense that it only depends on $\alpha$ and $b$. We also treat the case when the domain has multiple corners at the same point. Moreover, when $2b\leq \frac{1}{\alpha}$, we provide explicit constants for the upper and lower bounds.
We analyze the Boussinesq equation on the line with Schwartz initial data belonging to the physically relevant class of global solutions. In a recent paper, we determined ten main asymptotic sectors describing the large $(x,t)$-behavior of the solution, and for each of these sectors we provided the leading order asymptotics in the case when no solitons are present. In this paper, we give a formula valid in the asymptotic sector $x/t \in (1,M]$, where $M$ is a large positive constant, in the case when solitons are present. Combined with earlier results, this validates the soliton resolution conjecture for the Boussinesq equation everywhere in the $(x,t)$-plane except in a number of small transition zones.
In a recent paper, we showed that the large $(x,t)$ behavior of a class of physically relevant solutions of Boussinesq's equation for water waves is described by ten main asymptotic sectors. In the sector adjacent to the positive $x$-axis, referred to as Sector I, we stated without proof an exact expression for the leading asymptotic term together with an error estimate. Here we provide a proof of this asymptotic formula.
We consider the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term in the case when no solitons are present. In this paper, we derive an asymptotic formula in Sector IV, characterized by $\frac{x}{t}\in (\frac{1}{\sqrt{3}},1)$, in the case when solitons are present. In particular, our results provide an exact expression for the soliton-radiation interaction to leading order and a verification of the soliton resolution conjecture for the Boussinesq equation in Sector IV.
In a recent paper, we developed an inverse scattering approach to the Boussinesq equation in the case when no solitons are present. In this paper, we extend this approach to include solutions with solitons.
We construct a non-chiral conformal field theory (CFT) on the torus that accommodates a second quantization of the elliptic Calogero-Sutherland (eCS) model. We show that the CFT operator that provides this second quantization defines, at the same time, a quantum version of a soliton equation called the non-chiral intermediate long-wave (ncILW) equation. We also show that this CFT operator is a second quantization of a generalized eCS model which can describe arbitrary numbers of four different kinds of particles; we propose that these particles can be identified with solitons of the quantum ncILW equation.
We consider the Boussinesq equation on the line for a broad class of Schwartz initial data for which (i) no solitons are present, (ii) the spectral functions have generic behavior near $\pm 1$, and (iii) the solution exists globally. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term. In this paper, we give a proof for the formula corresponding to the sector $\frac{x}{t}\in (0,\frac{1}{\sqrt{3}})$.
It is well-known that each solution of the mKdV equation gives rise, via the Miura transformation, to a solution of the KdV equation. In this work, we show that a similar Miura-type transformation exists also for the ``good'' Boussinesq equation. This transformation maps solutions of a second-order equation to solutions of the fourth-order Boussinesq equation. Just like in the case of mKdV and KdV, the correspondence exists also at the level of the underlying Riemann--Hilbert problems and this is in fact how we construct the new transformation.
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.
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 B\"{a}cklund 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.
We consider the 4D fixed length lattice Higgs model with Wilson action for the gauge field and structure group $\mathbb{Z}_n$. We study Wilson line observables in the strong coupling regime and compute their asymptotic behavior with error estimates. Our analysis is based on a high-temperature representation of the lattice Higgs measure combined with Poisson approximation. We also give a short proof of the folklore result that Wilson line (and loop) expectations exhibit perimeter law decay whenever the Higgs field coupling constant is positive.
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.