Related papers: Long-time asymptotics of the KdV equation with del…
In this paper, the renowned Riemann-Hilbert method is employed to investigate the initial value problem of Tzitz\'eica equation on the line. Initially, our analysis focuses on elucidating the properties of two reflection coefficients, which…
In this article, we apply Deift-Zhou nonlinear steepest descent method to analyze the long-time asymptotic behavior of the solution for the discrete defocusing mKdV equation. This equation was proposed by Ablowitz and Ladik.
The long-time asymptotic behavior of the focusing nonlinear Schr\"odinger (NLS) equation on the line with symmetric nonzero boundary conditions at infinity is characterized by using the recently developed inverse scattering transform (IST)…
We investigate the soliton resolution and Painlev\'e asymptotics for the focusing Ablowitz-Ladik system with the initial data in a discrete weighted $\ell^2$ space. First, we establish the global well-posedness of this initial-value…
In this work, we study the Cauchy problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation with rapid attenuation of initial data. The basis Riemann-Hilbert problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is…
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…
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…
Consideration in this present paper is the long-time asymptotic of solutions to the derivative nonlinear Schr$\ddot{o}$dinger equation with the step-like initial value \begin{eqnarray} q(x,0)=q_{0}(x)=\begin{cases} \begin{split}…
We present a new and relatively elementary method for studying the solution of the initial-value problem for dispersive linear and integrable equations in the large-$t$ limit, based on a generalization of steepest descent techniques for…
The long-time asymptotic behavior of solutions to the focusing nonlinear Schr\"odinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the…
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction…
We consider dispersive shock wave to the focusing nonlinear Schr\"odinger equation generated by a discontinuous initial condition which is periodic or quasi-periodic on the left semi-axis and zero on the right semi-axis. As an initial…
This paper employs the Riemann-Hilbert problem to provide a comprehensive analysis of the asymptotic behavior of the high-genus Korteweg-de Vries soliton gases. It is demonstrated that the two-genus soliton gas is related to the two-phase…
This work investigates the long-time asymptotic behaviors of initial value problem for the good Boussinesq equation and the modified Boussinesq equation in Painlev\'{e} region. The Deift-Zhou steepest descent method is used to deform the…
In this work, we are devoted to study the Cauchy problem of the Camassa-Holm (CH) equation with weighted Sobolev initial data in space-time solitonic regions \begin{align*} m_t+2\kappa q_x+3qq_x=2q_xq_{xx}+qq_{xx},~~m=q-q_{xx}+\kappa,\\…
We study the asymptotic behavior of oscillatory Riemann-Hilbert problems arising in the AKNS hierarchy of integrable nonlinear PDE's. Our method is based on the Deift-Zhou nonlinear steepest descent method in which the given Riemann-Hilbert…
In this paper we study the asymptotics of solutions to the Korteweg--de Vries equation with steplike initial data, which lead to shock waves in the region between the asymptotically constant region and the soliton region, as $t \rightarrow…
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)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0,…
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}+\kappa u_{x}=0,…
In this paper, we obtain the long-time asymptotics of complex mKdV equation via Defit-Zhou method (Non-linear steepest descent method). The Cauchy problem of complex mKdV equation is transformed into the corresponding Riemann-Hilbert…