MR Author ID: 1360469
orcid.org/0000-0003-0348-877X
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Preprints

  1. Confluent hypergeometric kernel determinant on multiple large intervals
    with Lun Zhang and Zhengyang Zhao
    [arXiv] [Abstract]

    2. Abstract: 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.

  2. On the large-time asymptotics of the defocusing mKdV equation with step-like initial data
    [arXiv]

Publications in refereed journals

  1. Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions
    with Engui Fan and Zhaoyu Wang
    Communications in Mathematical Physics, 406 (2025), 181.
    [arXiv] [Journal] [Abstract]

    2. Abstract: 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.

  2. Soliton resolution and asymptotic stability of N-soliton solutions for the defocusing mKdV equation with a non-vanishing background
    with Engui Fan and Zechuan Zhang
    Physica D: Nonlinear Phenomena, 472 (2025), 134526.
    [arXiv] [Journal] [Abstract]

    3. Abstract: 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.

  3. Transient asymptotics of the modified Camassa-Holm equation
    with Yiling Yang and Lun Zhang
    Journal of the London Mathematical Society, 110 (2024), e12967.
    [arXiv] [Journal] [Abstract]

    4. Abstract: 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.

  4. On the Cauchy problem of defocusing mKdV equation with finite density initial data: long-time asymptotics in soliton-less regions
    with Engui Fan and Zechuan Zhang
    Journal of Differential Equations, 372 (2023), 55--122.
    [arXiv] [Journal] [Abstract]

    5. Abstract: 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 $\{(x,t): \vert \xi \vert < 6\}$ with $\xi:=x/t$. In the present paper, via the Riemann-Hilbert (RH) problem associated to the Cauchy problem, the long-time asymptotics in the soliton-less regions $\{(x,t): \vert \xi \vert>6, |\xi|=\mathcal{O}(1)\}$ 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.

  5. Large-time asymptotics to the focusing nonlocal modified Korteweg-de Vries equation with step-like boundary conditions
    with Engui Fan
    Studies in Applied Mathematics, 150 (2023), 1217--1273.
    [arXiv] [Journal] [Abstract]

    6. Abstract: We investigate the large-time asymptotics of solution for the Cauchy problem of the nonlinear focusing nonlocal 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 as $t\rightarrow-\infty$ 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.

  6. Riemann-Hilbert approach for multisoliton solutions of generalized coupled fourth-order nonlinear Schrödinger equations
    with Weiqi Peng and Shoufu Tian
    Mathematical Methods in the Applied Sciences, 43 (2020), 865--880.
    [Journal] [Abstract]

    7. Abstract: The main purpose of this work is to develop Riemann-Hilbert approach to obtain the soliton-solutions for generalized coupled fourth-order nonlinear Schr\"{o}dinger equations, which describe the simultaneous optical fiber. Starting from the spectral analysis of the Lax pair, a Riemann-Hilbert problem is set up. After solving the obtained Riemann-Hilbert problem with reflection-less case, we systematically derive multi-soliton solutions for the generalized coupled fourth-order nonlinear Schr\"{o}dinger equations. In addition, the localized structures and dynamic behaviors of one- and two- soliton solutions are shown by some graphic analysis.

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