arXiv subjects

J. Lenells

Publications and source records attributed to J. Lenells.

At least 19 recordsLinked to original sources

arXiv2026

A novel asymptotic technique for integrals involving the Hankel contour and the Bleistein asymptotic formula

Several important functions, including the gamma function, as well as several infinite sums, admit integral representations involving the Hankel contour. In addition, the large $t$ asymptotic analysis of several recently derived identities satisfied by the Riemann zeta function requires computing the asymptotic form of certain integrals which also involve the Hankel contour; these integrals depend on a real parameter, $\alpha$. A rigorous asymptotic technique is presented here for computing such integrals to all orders. For certain values of $\alpha$, the relevant formula, in addition to an asymptotic series of explicit terms, also contains a specific integral. It is shown that, remarkably, the leading order of this integral can be written in the form of the leading order of the Bleistein integral. The latter integral arises in the implementation of the classical steepest descent method in the case that the stationary point coincides with one of the boundary points of the integral under consideration.

math.CA↗

arXiv2025

Mixed phases in a Fermi--Hubbard model describing altermagnetism

We study an extension of the 2D Fermi--Hubbard model, which was recently introduced in [Das et al., Phys. Rev. Lett. 132, 263402 (2024)] and shown to describe altermagnetism that can be studied in cold atom systems. Using an updated Hartree--Fock method that can detect instabilities towards phase separation, we show that the model is in a mixed phase in large parts of the parameter regime at half-filling. We argue that the occurrence of a mixed phase is an indication of exotic physics which, in this model, occurs in parameter regimes accessible in cold atom experiments.

cond-mat.quant-gas↗

arXiv2025

Update of Hartree--Fock theory for Hubbard-like models

We show that the standard textbook description of (restricted) Hartree--Fock theory for (Fermi) Hubbard-like models is in need of an update, and we present such an update allowing us to correct basic and established results in the condensed matter physics literature that are qualitatively wrong. Our update amounts to adding a test which reliably checks the thermodynamic stability of solutions of Hartree--Fock equations. This stability test makes it possible to detect, by simple means and with certainty, regions in phase space where the model exhibits mixed phases where two conventional phases coexist and translation invariance is broken in complicated ways; in such a mixed phase, unconventional physics is to be expected. Our results show that mixed phases are ubiquitous in Hubbard-like models in arbitrary dimensions.

cond-mat.str-el↗

arXiv2024

Universality of mean-field antiferromagnetic order in an anisotropic 3D Hubbard model at half-filling

We study the 3D anisotropic Hubbard model on a cubic lattice with hopping parameter $t$ in the $x$- and $y$-directions and a possibly different hopping parameter $t_z$ in the $z$-direction; this model interpolates between the 2D and 3D Hubbard models corresponding to the limiting cases $t_z=0$ and $t_z=t$, respectively. We first derive all-order asymptotic expansions for the density of states. Using these expansions and units such that $t=1$, we analyze how the N\'eel temperature and the antiferromagnetic mean field depend on the coupling parameter, $U$, and on the hopping parameter $t_z$. We derive asymptotic formulas valid in the weak coupling regime, and we study in particular the transition from the three-dimensional to the two-dimensional model as $t_z \to 0$. It is found that the asymptotic formulas are qualitatively different for $t_z = 0$ (the two-dimensional case) and $t_z > 0$ (the case of nonzero hopping in the $z$-direction). Our results show that certain universality features of the three-dimensional Hubbard model are lost in the limit $t_z \to 0$ in which the three-dimensional model reduces to the two-dimensional model.

math-ph↗

arXiv2022

On Boussinesq's equation for water waves

A century and a half ago, J. Boussinesq derived an equation for the propagation of water waves in a channel. Despite the fundamental importance of this equation for a number of physical phenomena, mathematical results on it remain scarce. One reason for this is that the equation is ill-posed. In this paper, we establish several results on the Boussinesq equation. First, by solving the direct and inverse problems for an associated third-order spectral problem, we develop an Inverse Scattering Transform (IST) approach to the initial value problem. Using this approach, we establish a number of existence, uniqueness, and blow-up results. For example, the IST approach allows us to identify physically meaningful global solutions and to construct, for each $T > 0$, solutions that blow up exactly at time $T$. Our approach also yields an expression for the solution of the initial value problem for the Boussinesq equation in terms of the solution of a Riemann--Hilbert problem. By analyzing this Riemann--Hilbert problem, we arrive at asymptotic formulas for the solution. We identify ten main asymptotic sectors in the $(x,t)$-plane; in each of these sectors, we compute an exact expression for the leading asymptotic term together with a precise error estimate.

math.AP↗

arXiv2021

The implementation of the unified transform to the nonlinear Schr\"odinger equation with periodic initial conditions

The unified transform method (UTM) provides a novel approach to the analysis of initial-boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann-Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schr\"odinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial-boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial-boundary value problem on a finite interval with $x$-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle), belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg-de Vries (KdV) and modified KdV equations.

math.AP↗

arXiv2021

A new approach to integrable evolution equations on the circle

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schr\"odinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann-Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform.

nlin.SI↗

arXiv2020

The "good'' Boussinesq equation: long-time asymptotics

We consider the initial-value problem for the ``good'' Boussinesq equation on the line. Using inverse scattering techniques, the solution can be expressed in terms of the solution of a $3 \times 3$-matrix Riemann-Hilbert problem. We establish formulas for the long-time asymptotics of the solution by performing a Deift-Zhou steepest descent analysis of a regularized version of this Riemann-Hilbert problem.

math.AP↗

arXiv2020

Linearizable boundary value problems for the elliptic sine-Gordon and the elliptic Ernst equations

By employing a novel generalization of the inverse scattering transform method known as the unified transform or Fokas method, it can be shown that the solution of certain physically significant boundary value problems for the elliptic sine-Gordon equation, as well as for the elliptic version of the Ernst equation, can be expressed in terms of the solution of appropriate $2 \times 2$-matrix Riemann--Hilbert (RH) problems. These RH problems are defined in terms of certain functions, called spectral functions, which involve the given boundary conditions, but also unknown boundary values. For arbitrary boundary conditions, the determination of these unknown boundary values requires the analysis of a nonlinear Fredholm integral equation. However, there exist particular boundary conditions, called linearizable, for which it is possible to bypass this nonlinear step and to characterize the spectral functions directly in terms of the given boundary conditions. Here, we review the implementation of this effective procedure for the following linearizable boundary value problems: (a) the elliptic sine-Gordon equation in a semi-strip with zero Dirichlet boundary values on the unbounded sides and with constant Dirichlet boundary value on the bounded side; (b) the elliptic Ernst equation with boundary conditions corresponding to a uniformly rotating disk of dust; (c) the elliptic Ernst equation with boundary conditions corresponding to a disk rotating uniformly around a central black hole; (d) the elliptic Ernst equation with vanishing Neumann boundary values on a rotating disk.

math.AP↗

arXiv2017

Long-time asymptotics for the derivative nonlinear Schr\"odinger equation on the half-line

We derive asymptotic formulas for the solution of the derivative nonlinear Schr\"odinger equation on the half-line under the assumption that the initial and boundary values lie in the Schwartz class. The formulas clearly show the effect of the boundary on the solution. The approach is based on a nonlinear steepest descent analysis of an associated Riemann-Hilbert problem.

nlin.SI↗

arXiv2015

Long-time asymptotics for the Degasperis-Procesi equation on the half-line

We analyze the long-time asymptotics for the Degasperis--Procesi equation on the half-line. By applying nonlinear steepest descent techniques to an associated $3 \times 3$-matrix valued Riemann--Hilbert problem, we find an explicit formula for the leading order asymptotics of the solution in the similarity region in terms of the initial and boundary values.

math.AP↗

arXiv2014

The nonlinear Schr\"odinger equation with $t$-periodic data: I. Exact results

We consider the nonlinear Schr\"odinger equation on the half-line with a given Dirichlet (Neumann) boundary datum which for large $t$ tends to the periodic function $g_0^b(t)$ ($g_1^b(t)$). Assuming that the unknown Neumann (Dirichlet) boundary value tends for large $t$ to a periodic function $g_1^b(t)$ ($g_0^b(t)$), we derive an easily verifiable condition that the functions $g_0^b(t)$ and $g_1^b(t)$ must satisfy. Furthermore, we introduce two different methods, one based on the formulation of a Riemann-Hilbert problem, and one based on a perturbative approach, for constructing $g_1^b(t)$ ($g_0^b(t)$) in terms of $g_0^b(t)$ ($g_1^b(t)$).

math.AP↗

arXiv2014

The nonlinear Schr\"odinger equation with $t$-periodic data: II. Perturbative results

We consider the nonlinear Schr\"odinger equation on the half-line with a given Dirichlet boundary datum which for large $t$ tends to a periodic function. We assume that this function is sufficiently small, namely that it can be expressed in the form $\alpha g_0^b(t)$, where $\alpha$ is a small constant. Assuming that the Neumann boundary value tends for large $t$ to the periodic function $g_1^b(t)$, we show that $g_1^b(t)$ can be expressed in terms of a perturbation series in $\alpha$ which can be constructed explicitly to any desired order. As an illustration, we compute $g_1^b(t)$ to order $\alpha^8$ for the particular case that $g_0^b(t)$ is the sum of two exponentials. We also show that there exist particular functions $g_0^b(t)$ for which the above series can be summed up, and therefore for these functions $g_1^b(t)$ can be obtained in closed form. The simplest such function is $\exp(i\omega t)$, where $\omega$ is a real constant.

math.AP↗

arXiv2014

The Unified Transform for the modified Helmholtz equation in the exterior of a square

The Unified Transform provides a novel method for analyzing boundary value problems for linear and for integrable nonlinear PDEs. The numerical implementation of this method to linear elliptic PDEs formulated in the {\it interior} of a polygon has been investigated by several authors (see the article by Iserles, Smitheman, and one of the authors in this book). Here, we show that the Unified Transform also yields a novel numerical technique for computing the solution of linear elliptic PDEs in the {\it exterior} of a polygon. One of the advantages of this new technique is that it actually yields directly the scattering amplitude. Details are presented for the modified Helmholtz equation in the exterior of a square.

math.AP↗

arXiv2014

Perturbative and exact results on the Neumann value for the nonlinear Schr\"odinger equation on the half-line

The most challenging problem in the implementation of the so-called \textit{unified transform} to the analysis of the nonlinear Schr\"odinger equation on the half-line is the characterization of the unknown boundary value in terms of the given initial and boundary conditions. For the so-called \textit{linearizable} boundary conditions this problem can be solved explicitly. Furthermore, for non-linearizable boundary conditions which decay for large $t$, this problem can be largely bypassed in the sense that the unified transform yields useful asymptotic information for the large $t$ behavior of the solution. However, for the physically important case of periodic boundary conditions it is necessary to characterize the unknown boundary value. Here, we first present a perturbative scheme which can be used to compute explicitly the asymptotic form of the Neumann boundary value in terms of the given $\tau$-periodic Dirichlet datum to any given order in a perturbation expansion. We then discuss briefly an extension of the pioneering results of Boutet de Monvel and co-authors which suggests that if the Dirichlet datum belongs to a large class of particular $\tau$-periodic functions, which includes $\{a \exp(i \omega t) \, | \, a>0, \, \omega \geq a^2\}$, then the large $t$ behavior of the Neumann value is given by a $\tau$-periodic function which can be computed explicitly.

nlin.SI↗

arXiv2012

On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function

We present several formulae for the large $t$ asymptotics of the Riemann zeta function $\zeta(s)$, $s=\sigma+i t$, $0\leq \sigma \leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum $\sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $\sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $\Phi(u,v,\beta)$, $u\in \mathbb{C}$, $v\in \mathbb{C}$, $\beta \in \mathbb{R}$. Generalizing the methodology used in the study of $\zeta(s)$, we derive asymptotic formulae for $\Phi(u,v,\beta)$.

math.NT↗

arXiv2011

The Unified Method: I Non-Linearizable Problems on the Half-Line

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant, and on the computation of the large $k$ asymptotics of the eigenfunctions defining the relevant spectral functions.

math.AP↗

arXiv2011

The Unified Method: II NLS on the Half-Line with $t$-Periodic Boundary Conditions

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we first present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation and on the introduction of the so-called Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the spectral functions. We then concentrate on the physically significant case of $t$-periodic Dirichlet boundary data. After presenting certain heuristic arguments which suggest that the Neumann boundary values become periodic as $t\to\infty$, we show that for the case of the NLS with a sine-wave as Dirichlet data, the asymptotics of the Neumann boundary values can be computed explicitly at least up to third order in a perturbative expansion and indeed at least up to this order are asymptotically periodic.

math.AP↗