arXiv subjects

Engui Fan

Publications and source records attributed to Engui Fan.

At least 55 records · Page 3Linked to original sources

arXiv2022

Existence of global solutions to the nonlocal Schrödinger equation on the line

In this paper, we address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (nonlocal NLS) equation with the initial data $q_0(x)\in H^{1,1}(\R)$ with the $L^1(\R)$ small-norm assumption. We rigorously show that the spectral problem for the nonlocal NLS equation admits no eigenvalues or resonances, as well as Zhou vanishing lemma is effective under the $L^1(\R)$ small-norm assumption. With inverse scattering theory and the Riemann-Hilbert approach, we rigorously establish the bijectivity and Lipschitz continuous of the direct and inverse scattering map from the initial data to reflection coefficients.By using reconstruction formula and the Plemelj projection estimates of reflection coefficients,we further obtain the existence of the local solution and the priori estimates, which assure the existence of the global solution to the Cauchy problem for the nonlocal NLS equation.

math.AP↗

arXiv2022

The defocusing NLS equation with nonzero background: Large-time asymptotics in the solitonless region

We consider the Cauchy problem for the defocusing Schr$\ddot{\text{o}}$dinger (NLS) equation with a nonzero background $$\begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, \nonumber\\ &q(x,0)=q_0(x), \quad \lim_{x \to \pm \infty}q_0(x)=\pm 1. \end{align}$$ Recently, for the space-time region $|x/(2t)|<1$ which is a solitonic region without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the $N$-soliton solutions for the NLS equation by using the $\bar{\partial}$ generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time asymptotic expansion takes the form \begin{align} q(x,t)= T(\infty)^{-2} q^{sol,N}(x,t) + \mathcal{O}(t^{-1 }),\label{res1} \end{align} whose leading term is N-soliton and the second term $\mathcal{O}(t^{-1})$ is a residual error from a $\overline\partial$-equation. In this paper, we are interested in the large-time asymptotics in the space-time region $ |x/(2t)|>1$ which is outside the soliton region, but there will be two stationary points appearing on the jump contour $\mathbb{R}$. We found a asymptotic expansion that is different from (\ref{res1}) $$\begin{align} q(x,t)= e^{-iα(\infty)} \left(1 +t^{-1/2} h(x,t) \right)+\mathcal{O}\left(t^{-3/4}\right),\label{res2} \end{align}$$ whose leading term is a nonzero background, the second $t^{-1/2}$ order term is from continuous spectrum and the third term $\mathcal{O}(t^{-3/4})$ is a residual error from a $\overline\partial$-equation.The above two asymptotic results (\ref{res1}) and (\ref{res2}) imply that the region $ |x/(2t)|<1$ considered by Cuccagna and Jenkins is a fast decaying soliton solution region, while the region $ |x/(2t)|>1$ considered by us is a slow decaying nonzero background region.

math.AP↗

arXiv2022

On the long-time asymptotics of the modified Camassa-Holm equation with step-like initial data

We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with step-like initial data \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}=0, \quad m=u-u_{xx}, \nonumber \\ &u(x,0)=u_0(x)\to \left\{ \begin{array}{ll} A_1, &\ x\to+\infty,\\[5pt] A_2, &\ x\to-\infty, \end{array}\right.\nonumber \end{align} where $A_1$ and $A_2$ are two positive constants. Our main technical tool is the representation of the Cauchy problem with an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the step-like initial problem is characterized via the solution of a RH problem in the new scale $(y,t)$. We adopt double coordinates $(ξ, c)$ to divide the half-plane $\{ (ξ,c): ξ\in \mathbb{R}, \ c> 0, \ ξ=y/t\}$ into four asymptotic regions. Further using the Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution $u(y,t)$ in different space-time regions by the different choice of g-function. The corresponding leading asymptotic approximations are given with the slow/fast decay step-like background wave in genus-0 regions and elliptic waves in genus-2 regions. The second term of the asymptotics is characterized by Airy function or parabolic cylinder model. Their residual error order is $\mathcal{O}(t^{-1})$ or $\mathcal{O}(t^{-2})$ respectively.

nlin.SI↗

arXiv2022

Long-time asymptotic behavior for the Novikov equation in solitonic regions of space time

In this paper, we study the long time asymptotic behavior for the Cauchy problem of the Novikov equation with $3\times 3$ matrix spectral problem \begin{align} &u_{t}-u_{txx}+4 u_{x}=3uu_xu_{xx}+u^2u_{xxx}, \nonumber &u(x, 0)=u_{0}(x),\nonumber \end{align} where $u_0(x)$ $u_0(x)\rightarrow κ>0, \ x\rightarrow \pm \infty$ and $u_0(x)-κ$ is assumed in the Schwarz space. It is shown that the solution of the Cauchy problem can be characterized via a Riemann-Hilbert problem in a new scale $(y,t)$ with $$y=x-\int_{x}^{\infty}\left( (u-u_{xx}+1)^{2/3} -1\right) ds.$$ In different space-time solitonic regions of $ξ=y/t\in (-\infty,-1/8)\cup(1,+\infty) $ and $ξ\in(-1/8,1)$, we apply $\overline\partial$ steepest descent method to obtain the different long time asymptotic expansions of the solution $u(y,t)$. The corresponding residual error order is $\mathcal{O}(t^{-1+ρ})$ and $\mathcal{O}(t^{-3/4})$ respectively from a $\overline\partial$-equation. Our result implies that soliton resolution can be characterized with an $N(Λ)$-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the regions.

math-ph↗

arXiv2022

Long time asymptotic behavior for the nonlocal mKdV equation in space-time solitonic regions-II

We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions \begin{align*} &q_t(x,t)-6σq(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0, &q(x,0)=q_{0}(x),\ \ \lim_{x\to \pm\infty} q_{0}(x)=q_{\pm}, \end{align*} where $|q_{\pm}|=1$ and $q_{+}=δq_{-}$, $σδ=-1$. In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region $-6<ξ<6$ with $ξ=\frac{x}{t}$. In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for other solitonic regions $ξ<-6$ and $ξ>6$. Based on the Riemann-Hilbert problem of the the Cauchy problem, further using the $\bar{\partial}$ steepest descent method, we derive different long time asymptotic expansions of the solution $q(x,t)$ in above two different space-time solitonic regions. In the region $ξ<-6$, phase function $θ(z)$ has four stationary phase points on the $\mathbb{R}$. Correspondingly, $q(x,t)$ can be characterized with an $\mathcal{N}(Λ)$-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function ${\rm Im}ν(ζ_i)$. In the region $ξ>6$, phase function $θ(z)$ has four stationary phase points on $i\mathbb{R}$, the corresponding asymptotic approximations can be characterized with an $\mathcal{N}(Λ)$-soliton with diverse residual error order $\mathcal{O}(t^{-1})$.

math.AP↗

arXiv2022

$L^2$ Sobolev space bijectivity of the scattering-inverse scattering transforms related to defocusing Ablowitz-Ladik systems

In this paper, we establish $L^2$-Sobolev space bijectivity of the inverse scattering transform related to the defocusing Ablowitz-Ladik system. On the one hand, in the direct problem, based on the spectral problem, we establish the reflection coefficient and the corespondent Riemann-Hilbert problem. And we also prove that if the potential belongs to $l^{2,k}$ space, then the reflection coefficient belongs to $H^k_θ(Σ)$. On the other hand, in the inverse problem, based on the Riemann-Hilbert problem, we obtain the corespondent reconstructed formula and recover potentials from reflection coefficients. And we also confirm that if reflection coefficients are in $H^k_θ(Σ)$, then we show that potentials also belong to $l^{2,k}$. This study also confirm that for the initial-valued problem of defocusing Ablowitz-Ladik equations, it the initial potential belongs to $l^{2,k}$ and satisfying $\parallel q\parallel_\infty<1$, then the solution for $t\ne0$ also belongs to $l^{2,k}$.

math.AP↗

arXiv2022

Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-time

We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation $$ u_{xt}+αβ^2u-2iαβu_x-αu_{xx}-iαβ^2|u|^2u_x=0 \label{cs} $$ with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at $k=0$. (ii) four stationary phase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial cone $$\mathcal{C}\left(x_{1}, x_{2}, v_{1}, v_{2}\right)=\left\{(x, t) \in \mathbb{R}^{2} | x=x_{0}+v t, x_{0} \in\left[x_{1}, x_{2}\right], v \in\left[v_{1}, v_{2}\right]\right\},$$ the long-time asymptotic behavior of the solution $u(x,t)$ for the focusing FL equation can be characterized with an $N(\mathcal{I})$-soliton on discrete spectrums and a leading order term $\mathcal{O}(|t|^{-1/2})$ on continuous spectrum up to a residual error order $\mathcal{O}(|t|^{-3/4})$. The main tool is the $\overline{\partial}$ nonlinear steepest descent method and the $\overline{\partial}$-analysis.

math.AP↗

arXiv2022

Painleve-type asymptotics for the defocusing Hirota equation in transition region

We consider the Cauchy problem for the classical Hirota equation on the line with decaying initial data. Based on the spectral analysis of the Lax pair of the Hirota equation, we first expressed the solution of the Cauchy problem in terms of the solution of a Riemann-Hilbert problem. Further we apply nonlinear steepest descent analysis to obtain the long-time asymptotics of the solution in the critical transition region $|\frac{x}{t} - \frac{α^2}{3β}|t^{2/3} \leq M$, $M$ is a positive constant. Our result shows that the long time asymptotics of the Hirota equation can be expressed in terms of the solution of Painlevé $\mathrm{II}$ equation. Keywords: Hirota equation, steepest descent method, Painlevé $\mathrm{II}$ equation, long-time asymptotics.

math-ph↗

arXiv2021

Long time asymptotics for the nonlocal mKdV equation with finite density initial data

In this paper, we consider the Cauchy problem for an integrable real nonlocal (also called reverse-space-time) mKdV equation with nonzero boundary conditions \begin{align*} &q_t(x,t)-6σq(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0, &q(x,0)=q_{0}(x),\lim_{x\to \pm\infty} q_{0}(x)=q_{\pm}, \end{align*} where $|q_{\pm}|=1$ and $q_{+}=δq_{-}$, $σδ=-1$. Based on the spectral analysis of the Lax pair, we express the solution of the Cauchy problem of the nonlocal mKdV equation in terms of a Riemann-Hilbert problem. In a fixed space-time solitonic region $-6<x/t<6$, we apply $\bar{\partial}$-steepest descent method to analyze the long-time asymptotic behavior of the solution $q(x,t)$. We find that the long time asymptotic behavior of $q(x,t)$ can be characterized with an $N(Λ)$-soliton on discrete spectrum and leading order term $\mathcal{O}(t^{-1/2})$ on continuous spectrum up to an residual error order $\mathcal{O}(t^{-1})$.

math.AP↗

arXiv2021

Long time and Painleve-type asymptotics for the Sasa-Satsuma equation in solitonic space time regions

The Sasa-Satsuma equation with $3 \times 3 $ Lax representation is one of the integrable extensions of the nonlinear Schrödinger equation. In this paper, we consider the Cauchy problem of the Sasa-Satsuma equation with generic decaying initial data. Based on the Rieamnn-Hilbert problem characterization for the Cauchy problem and the $\overline{\partial}$-nonlinear steepest descent method, we find qualitatively different long time asymptotic forms for the Sasa-Satsuma equation in three solitonic space-time regions: (1)\ For the region $x<0, |x/t|=\mathcal{O}(1)$, the long time asymptotic is given by $$q(x,t)=u_{sol}(x,t| σ_{d}(\mathcal{I})) + t^{-1/2} h + \mathcal{O} (t^{-3/4}). $$ in which the leading term is $N(I)$ solitons, the second term the second $t^{-1/2}$ order term is soliton-radiation interactions and the third term is a residual error from a $\overline\partial$ equation. (2)\ For the region $ x>0, |x/t|=\mathcal{O}(1)$, the long time asymptotic is given by $$ u(x,t)= u_{sol}(x,t| σ_{d}(\mathcal{I})) + \mathcal{O}(t^{-1}).$$ in which the leading term is $N(I)$ solitons, the second term is a residual error from a $\overline\partial$ equation. (3) \ For the region $ |x/t^{1/3}|=\mathcal{O}(1)$, the Painleve asymptotic is found by $$ u(x,t)= \frac{1}{t^{1/3}} u_{P} \left(\frac{x}{t^{1/3}} \right) + \mathcal{O} \left(t^{2/(3p)-1/2} \right), \qquad 4<p < \infty.$$ in which the leading term is a solution to a modified Painleve $\mathrm{II}$ equation, the second term is a residual error from a $\overline\partial$ equation.

math.AP↗

arXiv2021

On the asymptotic stability of $N$-soliton solutions of the three-wave resonant interaction equation

The three-wave resonant interaction (three-wave) equation not only possesses $3\times 3$ matrix spectral problem, but also being absence of stationary phase points, which give rise to difficulty on the asymptotic analysis with stationary phase method or classical Deift-Zhou steepest descent method. In this paper, we study the long time asymptotics and asymptotic stability of $N$-soliton solutions of the initial value problem for the three-wave equation in the solitonic region \begin{align} &p_{ij,t}-n_{ij}p_{ij,x}+\sum_{k=1}^{3}(n_{kj}-n_{ik})p_{ik}p_{kj}=0, &p_{ij}(x, 0)=p_{ij,0}(x), \quad x \in \mathbb{R},\ t>0,\ i,j,k=1,2,3, \nonumber &for\ i\neq j,\ p_{ij}=-\bar{p}_{ji}, \ n_{ij}=-n_{ji}, \end{align} where $n_{ij}$ are constants. The study makes crucial use of the inverse scattering transform as well as of the $\overline\partial$ generalization of Deift-Zhou steepest descent method for oscillatory Riemann-Hilbert (RH) problems. Based on the spectral analysis of the Lax pair associated with the three-wave equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a RH problem. Further we derive the leading order approximation to the solution $p_{ij}(x, t)$ for the three-wave equation in the solitonic region of any fixed space-time cone. The asymptotic expansion can be characterized with an $N(I)$-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the region; the residual error order $\mathcal{O}(t^{-1})$ from a $\overline\partial$ equation. Our results provide a verification of the soliton resolution conjecture and asymptotic stability of N-soliton solutions for three-wave equation.

math.AP↗

arXiv2021

On asymptotic approximation of the modified Camassa-Holm equation in different space-time solitonic regions

In this paper, we study the long time asymptotic behavior for the initial value problem of the modified Camassa-Holm (mCH) equation in the solitonic region \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}+κu_{x}=0, \quad m=u-u_{x x}, \nonumber &u(x, 0)=u_{0}(x),\nonumber \end{align} where $κ$ is a positive constant. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a Riemann-Hilbert (RH) problem. Further using the $\overline\partial$ generalization of Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution $u(x,t)$ in different space-time solitonic region of $x/t$. These asymptotic approximations can be characterized with an $N(Λ)$-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the region with diverse residual error order from $\overline\partial$ equation: $\mathcal{O}(|t|^{-1+2ρ})$ for $ξ=\frac{y}{t}\in(-\infty,-0.25)\cup(2,+\infty)$ and $\mathcal{O}(|t|^{-3/4})$ for $ξ=\frac{y}{t}\in(-0.25,2)$. Our results also confirm the soliton resolution conjecture and asymptotically stability of N-soliton solutions for the mCH equation.

math.AP↗

arXiv2021

Long time asymptotics for the focusing nonlinear Schrödinger equation in the solitonic region with the presence of high-order discrete spectrum

In this paper, we use the $\bar{\partial}$ steepest descent method to study the initial value problem for focusing nonlinear Schrödinger (fNLS) equation with non-generic weighted Sobolev initial data that allows for the presence of high-order discrete spectrum. More precisely, we shall characterize the properties of the eigenfunctions and scattering coefficients in the presence of high-order poles; further we formulate an appropriate enlarged RH problem; after a series of deformations, the RH problem is transformed into a solvable model. Finally, we obtain the asymptotic expansion of the solution of the fNLS equation in any fixed space-time cone: %as $t \to \infty$, \begin{equation*} \mathcal{S}(x_1,x_2,v_1,v_2):=\left\lbrace (x,t)\in \mathbb{R}^2: x=x_0+vt, \ x_0\in[x_1,x_2]\text{, }v\in[v_1,v_2]\right\rbrace. \end{equation*} Observing the result indicates that the solution of fNLS equation in this case satisfies the soliton resolution conjecture. The leading order term of this solution includes a high-order pole-soliton whose parameters are affected by soliton-soliton interactions through the cone and soliton-radiation interactions on continuous spectrum. The error term of this result is up to $\mathcal{O}(t^{-3/4})$ which comes from the corresponding $\bar{\partial}$ equation.

math.AP↗

arXiv2021

Long-time asymptotic behavior of the nonlocal nonlinear Schrödinger equation with initial potential in weighted sobolev space

In this paper, we are going to investigate Cauchy problem for nonlocal nonlinear Schrödinger equation with the initial potential $q_0(x)$ in weighted sobolev space $H^{1,1}(\mathbb{R})$, \begin{align*} iq_t(x,t)&+q_{xx}(x,t)+2σq^2(x,t)\bar q(-x,t)=0,\quadσ=\pm1,\\ q(x,0)&=q_0(x). \end{align*} We show that the solution can be represented by the solution of a Riemann-Hilbert problem (RH problem), and assuming no discrete spectrum, we majorly apply $\bar\partial$-steepest cescent descent method on analyzing the long-time asymptotic behavior of it.

math.AP↗

arXiv2021

Long time asymptotic behavior for the derivative Schrödinger equation with nonzero boundary conditions

In this paper, we apply $\overline\partial$ steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with nonzero boundary conditions \begin{align} &iq_{t}+q_{xx}+iσ(|q|^2q)_{x}=0,\\ & (x,0) = q_0(x), \quad\lim_{x\to\pm\infty} q_0(x) = q_\pm,\end{align} where $|q_\pm|=1$. Based on the spectral analysis of the Lax pair, we express the solution of the derivative nonlinear Schrödinger equation in terms of solutions of a Riemann-Hilbert problem.In a fixed space-time solitonic region $-3<x/t<-1$, we compute the long time asymptotic expansion of the solution $q(x,t)$,which implies soliton resolution conjecture and can be characterized with an $N(Λ)$-soliton whose parameters are modulated bya sum of localized soliton-soliton interactions as one moves through the region; the residual error order $\mathcal{O}( t^{-3/4})$ from a $\overline\partial$ equation.

nlin.SI↗

arXiv2020

Long-time asymptotic behavior of a mixed schrödinger equation with weighted Sobolev initial data

We apply $\bar{\partial}$ steepest descent method to obtain sharp asymptotics for a mixed schrödinger equation $$ q_t+iq_{xx}-ia (\vert q \vert^2q)_x -2b^2\vert q \vert^2q=0,$$ $$q(x,t=0)=q_0(x),$$ under essentially minimal regularity assumptions on initial data in a weighted Sobolev space $q_0(x) \in H^{2,2}(\mathbb{R})$. In the asymptotic expression, the leading order term $\mathcal{O}(t^{-1/2})$ comes from dispersive part $q_t+iq_{xx}$ and the error order $\mathcal{O}(t^{-3/4})$ from a $\overline\partial$ equation

math.AP↗

arXiv2020

Soliton Resolution for the Short-pluse Equation

In this paper, we study the Cauchy problem for the focusing nonlinear short-pluse equation by using $\overline\partial$ steepest descent method. \begin{align} &u_{xt}=u+\frac{1}{6}(u^3)_{xx}, \nonumber\\ &u(x,0)=u_0(x)\in H^{1,1}(R),\nonumber \end{align} where $H^{1,1}(R)$ is a weighted Sobolev space. Because the spectral variable z is the same order in the WKI-type Lax pair, we construct the solution of SP equation in the new scale $(y,t)$, whereas the original scale $(x,t)$ is given in terms of functions in the new scale and the solution of Riemann-Hilbert problem. In any fixed space-time cone of the new scale $(y,t)$ which stratify that $v_1\leq v_1 \in R^-$ and $ξ=\frac{y}{t}<0$, \begin{equation} C(y_1,y_2,v_1,v_2) = \left\lbrace (y,t) \in R^2|y=y_0+vt, y_0 \in[y_1,y_2]\text{, } v\in[v_1,v_2]\right\rbrace, \nonumber \end{equation} we compute the long time asymptotic expansion of the solution $u(x,t)$, which prove soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an $N(I)$-soliton whose parameters are modulated by a sum of localied soliton-soliton interactions as one moves through the cone; the second $t^{-1/2}$ order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order $\mathcal{O}(|t|^{-1})$ from a $\overline\partial$ equation. Our results also show that soliton solutions of short-pluse equation are asymptotically stable.

nlin.SI↗

arXiv2020

Critical edge behavior in the singularly perturbed Pollaczek-Jacobi type unitary ensemble

In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight $$w_{p_J2}(x,t)=e^{-\frac{t}{x(1-x)}}x^α(1-x)^β, $$ where $t \ge 0$, $α>0$, $β>0$ and $x \in [0,1].$ Our main results obtained here include two aspects: { I. Strong asymptotics:} We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval $(0,1)$ and outside of interval $\mathbb{C}\backslash (0,1)$, respectively; Due to the effect of $\frac{t}{x(1-x)}$ for varying $t$, different asymptotic behaviors at the hard edge $0$ and $1$ were found with different scaling schemes. Specifically, the uniform asymptotic behavior can be expressed as a Airy function in the neighborhood of point $1$ as $ζ= 2n^2t \to \infty, n\to \infty$, while it is given by a Bessel function as $ζ\to 0, n \to \infty$. { II. Universality:} We respectively calculate the limit of the eigenvalue correlation kernel in the bulk of the spectrum and at the both side of hard edge, which will involve a $ψ$-functions associated with a particular Painlev$\acute{e}$ \uppercase\expandafter{\romannumeral3} equation near $x=\pm 1$. Further, we also prove the $ψ$-funcation can be approximated by a Bessel kernel as $ζ\to 0$ compared with a Airy kernel as $ζ\to \infty$. Our analysis is based on the Deift-Zhou nonlinear steepest descent method for the Riemann-Hilbert problems.

math.CA↗