arXiv subjects

Taiyang Xu

Publications and source records attributed to Taiyang Xu.

8 recordsLinked to original sources

arXiv2026

Moment generating function of the tacnode process

The tacnode process is a universal determinantal point process arising in non-intersecting particle systems and random tiling models. In this paper, we study the generating function for the counting functions of the tacnode process on a union of $m$ intervals, $m\in\mathbb{N}^{+}$. Our first result provides an integral representation for the $m$-point generating function in terms of the Hamiltonian governing a system of $8m+4$ coupled differential equations. Combined with several differential identities for this Hamiltonian, the representation yields the large gap asymptotics, up to and including the constant term. As further applications, we obtain asymptotic formulae for the expectations, variances, and covariances of the counting functions, and establish a central limit theorem for their joint fluctuations. These results extend the previously known $1$-point theory for the tacnode process to the multi-interval setting with multiple discontinuities.

math-ph↗

arXiv2025

Confluent hypergeometric kernel determinant on multiple large intervals

The confluent hypergeometric point process represents a universality class which arises in a variety of different but related areas. It particularly describes the local statistics of eigenvalues in the bulk of spectrum near a Fisher-Hartwig singular point for a broad class of unitary ensembles. It is the aim of this work to investigate large gap asymptotics of this process over a union of disjoint intervals $\cup_{j=0}^{n}(sa_j,sb_j)$, where $a_0<b_0<\dots<a_m<0<b_m<\dots<a_n<b_n$ for some $0\leq m \leq n$. As $s\to +\infty$, we establish a general asymptotic formula up to and including the oscillatory term of order $1$, which involves a $\theta$-functions-combination integral along a linear flow on an $n$-dimensional torus. If the linear flow has ``good Diophantine properties'' or the ergodic properties, we further improve the error estimate or the leading term for the asymptotics of the integral. These results can be combined for the case $n=1$, which lead to a precise large gap asymptotics up to an undetermined constant.

math-ph↗

arXiv2023

Transient asymptotics of the modified Camassa-Holm equation

We investigate long time asymptotics of the modified Camassa-Holm equation in three transition zones under a nonzero background. The first transition zone lies between the soliton region and the first oscillatory region, the second one lies between the second oscillatory region and the fast decay region, and possibly, the third one, namely, the collisionless shock region, that bridges the first transition region and the first oscillatory region. Under a low regularity condition on the initial data, we obtain Painlev\'e-type asymptotic formulas in the first two transition regions, while the transient asymptotics in the third region involves the Jacobi theta function. We establish our results by performing a $\bar{\partial}$ nonlinear steepest descent analysis to the associated Riemann-Hilbert problem.

math.AP↗

arXiv2023

On the Cauchy problem of defocusing mKdV equation with finite density initial data: long time asymptotics in soliton-less regions

We investigate the long-time asymptotics for the solutions to the Cauchy problem of defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data. The present paper is the subsequent work of our previous paper [arXiv:2108.03650], which gives the soliton resolution for the defocusing mKdV equation in the central asymptotic sector $\{(x,t): \vert ξ\vert<6\}$ with $ξ:=x/t$. In the present paper, via the Riemann-Hilbert (RH) problem associated to the Cauchy problem, the long-time asymptotics in the soliton-less regions $\{(x,t): \vert ξ\vert>6, |ξ|=\mathcal{O}(1)\}$ for the defocusing mKdV equation are further obtained. It is shown that the leading term of the asymptotics are in compatible with the ``background solution'' and the error terms are derived via rigorous analysis.

math.AP↗

arXiv2023

Painlev\'e transcendents in the defocusing mKdV equation with non-zero boundary conditions

We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with non-zero boundary conditions \begin{align} &q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \nonumber &q(x,0)=q_{0}(x)\to \pm 1, \ \ x\rightarrow\pm\infty, \nonumber \end{align} which can be characterized using a Riemann-Hilbert problem through the inverse scattering transform. Using the $\bar\partial$-generalization of the Deift-Zhou nonlinear steepest descent approach, combined with the double scaling limit technique, we obtain the long-time asymptotics of the solution of the Cauchy problem for the defocusing mKdV equation in the transition region $|x/t+6|t^{2/3}< C$ with $C>0$. The asymptotics can be expressed in terms of the solution of the second Painlev\'{e} transcendent.

math-ph↗

arXiv2022

Large-time asymptotics to the focusing nonlocal modified Kortweg-de Vries equation with step-like boundary conditions

We investigate the large-time asymptotics of solution for the Cauchy problem of the nonlocal focusing modified Kortweg-de Vries (MKdV) equation with step-like initial data, i.e., $u_0(x)\rightarrow 0$ as $x\rightarrow-\infty$, $u_0(x)\rightarrow A$ as $x\rightarrow+\infty$,where $A$ is an arbitrary positive real number. We firstly develop the direct scattering theory to establish the basic Riemann-Hilbert (RH) problem associated with step-like initial data. Thanks to the symmetries $x\rightarrow-x$, $t\rightarrow-t$ of nonlocal MKdV equation, we investigate the asymptotics for $t\rightarrow-\infty$ and $t\rightarrow+\infty$ respectively. Our main technique is to use the steepest descent analysis to deform the original matrix-valued RH problem to corresponded regular RH problem, which could be explicitly solved. Finally we obtain the different large-time asymptotic behaviors of the solution of the Cauchy problem for focusing nonlocal MKdV equation in different space-time sectors $\mathcal{R}_{I}$, $\mathcal{R}_{II}$, $\mathcal{R}_{III}$ and $\mathcal{R}_{IV}$ on the whole $(x,t)$-plane.

math-ph↗

arXiv2022

On the large-time asymptotics of the defocusing mKdV equation with step-like initial data

We study the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with step-like initial data approaching nonzero constants $c_l$ and $c_r$ as $x \to -\infty$ and $x\to+\infty$, respectively. Assuming $c_l>c_r>0$, the solution exhibits a rarefaction wave structure. We first develop the inverse scattering transform for the solution satisfying these step-like boundary conditions. Using the associated scattering data, we prove that there exists a unique global solution of the Cauchy problem and characterize it in terms of a Riemann-Hilbert (RH) problem. By applying the nonlinear steepest descent method to this RH problem, we rigorously obtain large-time asymptotics of rarefaction wave solution in three distinct space-time regions, each characterized by a different leading order behavior. They are: (I) a left-field region where the solution approaches the left background constant, modulo a small oscillatory correction, (II) a central region where the solution exhibits a slowly varying profile that transitions between the two constants, and (III) a right-field region where the solution tends to the right background constant, up to an algebraically small correction. Rigorous derivations of the leading terms, sub-leading terms as well as the error bounds are presented.

math.AP↗

arXiv2021

Soliton resolution and asymptotic stability of $N$-soliton solutions for the defocusing mKdV equation with a non-vanishing background

We analytically study the large-time asymptotics of the solution of the defocusing modified Korteweg-de Vries (mKdV) equation under a symmetric non-vanishing background, which supports the emergence of solitons. It is demonstrated that the asymptotic expansion of the solution at the large time could verify the renowned soliton resolution conjecture. Moreover, the asymptotic stability of $N$-soliton solution is also exhibited in the present work. We establish our results by performing a $\bar{\partial}$-nonlinear steepest descent analysis to the associated Riemann-Hilbert (RH) problem.

math.AP↗