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Anne Boutet de Monvel

Publications and source records attributed to Anne Boutet de Monvel.

At least 19 recordsLinked to original sources
arXiv2025

The periodic Camassa-Holm equation by the Riemann-Hilbert problem approach

This work addresses the development of the Riemann-Hilbert problem (RHP) formalism (the Fokas method) for the Camassa-Holm equation under periodic boundary conditions. Particularly, we present a representation of the solution to this problem in terms of the solution of the associated Riemann-Hilbert problem, the data for which are determined by the initial data for the problem in terms of the associated spectral functions.

math.AP↗
arXiv2025

Limiting distributions of ergodic continuous-time quantum walks on periodic graphs

In this expository note, we study several families of periodic graphs which satisfy a sufficient condition for the ergodicity of the associated continuous-time quantum walk. For these graphs, we compute the limiting distribution of the walk explicitly. We uncover interesting behavior where in some families, the walk is ergodic in both horizontal and sectional directions, while in others, ergodicity only holds in the horizontal (large N) direction. We compare this to the limiting distribution of classical random walks on the same graphs.

math-ph↗
arXiv2023

Ergodic theorems for continuous-time quantum walks on crystal lattices and the torus

We give several quantum dynamical analogs of the classical Kronecker-Weyl theorem, which says that the trajectory of free motion on the torus along almost every direction tends to equidistribute. As a quantum analog, we study the quantum walk $\exp(-i t Δ) ψ$ starting from a localized initial state $ψ$. Then the flow will be ergodic if this evolved state becomes equidistributed as time goes on. We prove that this is indeed the case for evolutions on the flat torus, provided we start from a point mass, and we prove discrete analogs of this result for crystal lattices. On some periodic graphs, the mass spreads out non-uniformly, on others it stays localized. Finally, we give examples of quantum evolutions on the sphere which do not equidistribute.

math-ph↗
arXiv2022

Focusing nonlocal nonlinear Schrödinger equation with asymmetric boundary conditions: large-time behavior

We consider the focusing integrable nonlocal nonlinear Schrödinger equation \[\mathrm{i}q_{t}(x,t)+q_{xx}(x,t)+2q^{2}(x,t)\bar{q}(-x,t)=0\] with asymmetric nonzero boundary conditions: $q(x,t)\to\pm A\mathrm{e}^{-2\mathrm{i}A^2t}$ as $x\to\pm\infty$, where $A>0$ is an arbitrary constant. The goal of this work is to study the asymptotics of the solution of the initial value problem for this equation as $t\to+\infty$. For a class of initial values we show that there exist three qualitatively different asymptotic zones in the $(x,t)$ plane. Namely, there are regions where the parameters are modulated (being dependent on the ratio $x/t$) and a central region, where the parameters are unmodulated. This asymptotic picture is reminiscent of that for the defocusing classical nonlinear Schrödinger equation, but with some important differences. In particular, the absolute value of the solution in all three regions depends on details of the initial data.

math.AP↗
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

Ballistic transport in periodic and random media

We prove ballistic transport of all orders, that is, $\lVert x^m\mathrm{e}^{-\mathrm{i}tH}ψ\rVert\asymp t^m$, for the following models: the adjacency matrix on $\mathbb{Z}^d$, the Laplace operator on $\mathbb{R}^d$, periodic Schrödinger operators on $\mathbb{R}^d$, and discrete periodic Schrödinger operators on periodic graphs. In all cases we give the exact expression of the limit of $\lVert x^m\mathrm{e}^{-\mathrm{i}tH}ψ\rVert/t^m$ as $t\to+\infty$. We then move to universal covers of finite graphs (these are infinite trees) and prove ballistic transport in mean when the potential is lifted naturally, giving a periodic model, and when the tree is endowed with random i.i.d.\ potential, giving an Anderson model. The limiting distributions are then discussed, enriching the transport theory. Some general upper bounds are detailed in the appendix.

math-ph↗
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↗
arXiv2020

The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem

This paper deals with the Cauchy problem for the modified Camassa-Holm (mCH) equation \begin{alignat*}{4} &m_t+\left((u^2-u_x^2)m\right)_x=0,&\quad&m:= u-u_{xx},&\quad&t>0,&\;&-\infty 0$), where the leading asymptotic term of the deviation of the solution from the background is nontrivial: this term is given by modulated (with parameters depending on $\frac{x}{t}$), decaying (as $t^{-1/2}$) trigonometric oscillations.

math.AP↗
arXiv2019

A Riemann-Hilbert approach to the modified Camassa-Holm equation with nonzero boundary conditions

The paper aims at developing the Riemann-Hilbert problem approach to the modified Camassa-Holm (mCH) equation in the case when the solution is assumed to approach a non-zero constant at the both infinities of the space variable. In this case, the spectral problem for the associated Lax pair equation has a continuous spectrum, which allows formulating the inverse spectral problem as a Riemann-Hilbert factorization problem with jump conditions across the real axis. We obtain a representation for the solution of the Cauchy problem for the mCH equation and also a description of certain soliton-type solutions, both regular and non-regular.

math-ph↗
arXiv2018

Oscillatory behavior of large eigenvalues in quantum Rabi models

We investigate the large $n$ asymptotics of the $n$-th eigenvalue for a class of unbounded self-adjoint operators defined by infinite Jacobi matrices with discrete spectrum. In the case of the quantum Rabi model we obtain the first three terms of the asymptotics which determine the parameters of the model. This paper is based on our previous paper [5] that it completes and improves.

math-ph↗
arXiv2016

A Riemann-Hilbert Approach for the Novikov Equation

We develop the inverse scattering transform method for the Novikov equation $u_t-u_{txx}+4u^2u_x=3u u_xu_{xx}+u^2u_{xxx}$ considered on the line $x\in(-\infty,\infty)$ in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann-Hilbert (RH) problem, which in this case is a $3\times 3$ matrix problem. The structure of this RH problem shares many common features with the case of the Degasperis-Procesi (DP) equation having quadratic nonlinear terms (see [Boutet de Monvel A., Shepelsky D., Nonlinearity 26 (2013), 2081-2107, arXiv:1107.5995]) and thus the Novikov equation can be viewed as a "modified DP equation", in analogy with the relationship between the Korteweg-de Vries (KdV) equation and the modified Korteweg-de Vries (mKdV) equation. We present parametric formulas giving the solution of the Cauchy problem for the Novikov equation in terms of the solution of the RH problem and discuss the possibilities to use the developed formalism for further studying of the Novikov equation.

nlin.SI↗
arXiv2016

Asymptotic behavior of large eigenvalues of Jaynes-Cummings type models

We consider a class of unbounded self-adjoint operators including the Hamiltonian of the Jaynes-Cummings model without the rotating-wave approximation (RWA). The corresponding operators are defined by infinite Jacobi matrices with discrete spectrum. Our purpose is to give the asymptotic behavior of large eigenvalues.

math-ph↗
arXiv2016

The short pulse equation by a Riemann-Hilbert approach

We develop a Riemann-Hilbert approach to the inverse scattering transform method for the short pulse (SP) equation $u_{xt}=u+\frac{1}{6}(u^3)_{xx}$ with zero boundary conditions (as $|x|\to\infty$). This approach is directly applied to the Lax pair for the SP equation. It allows us to give a parametric representation of the solution to the Cauchy problem. This representation is then used for studying the long-time behavior of the solution as well as for retrieving the soliton solutions. Finally, the analysis of the long-time behavior allows us to formulate, in spectral terms, a sufficient condition for the wave breaking.

nlin.SI↗
arXiv2013

The Ostrovsky-Vakhnenko equation by a Riemann-Hilbert approach

We present an inverse scattering transform approach for the equation $u_{txx}-3u_x+3u_xu_{xx}+uu_{xxx}=0$. This equation can be viewed as the short wave model for the Degasperis-Procesi equation or the differentiated Ostrovsky-Vakhnenko equation. The approach is based on an associated Riemann-Hilbert problem, which allows us to give a representation for the classical (smooth) solution, to get the principal term of its long time asymptotics, and also to describe loop soliton solutions.

nlin.SI↗
arXiv2013

New Applications of Quantum Algebraically Integrable Systems in Fluid Dynamics

The rational quantum algebraically integrable systems are non-trivial generalizations of Laplacian operators to the case of elliptic operators with variable coefficients. We study corresponding extensions of Laplacian growth connected with algebraically integrable systems, describing viscous free-boundary flows in non-homogenous media. We introduce a class of planar flows related with application of Adler-Moser polynomials and construct solutions for higher-dimensional cases, where the conformal mapping technique is unavailable.

nlin.SI↗