Related papers: Long-time asymptotics for the integrable nonlocal …

The methodology of the Riemann-Hilbert (RH) factorisation approach for Lax-pair isospectral deformations is used to derive, in the solitonless sector, the leading-order asymptotics as $t \to \pm \infty$ $(x/t \sim \mathcal{O}(1))$ of…

In this paper, we consider the Cauchy problem for the modified NLS equation. Using nonlinear steepest descent method and combining the Dbar-analysis, we show that inside any fixed cone, the long time asymptotic behavior of the solution for…

We investigate the Cauchy problem for the focusing nonlinear Schr\"odinger (NLS) equation \begin{equation} iq_t(x,t)+q_{xx}(x,t)+2|q(x,t)|^2q(x,t)=0,\quad x\in\mathbb{R},\quad t\ge0,\nonumber \end{equation} subject to initial data $ q(x,0)$…

In this work, we consider the long-time asymptotics for the Cauchy problem of a fourth-order dispersive nonlinear Schr\"{o}dinger equation with nonzero boundary conditions at infinity. Firstly, in order to construct the basic…

We investigate the Cauchy problem of a new higher-order nonlinear Schr\"{o}dinger equation (NHNSE) with weighted Sobolev initial data which is derived by ourselves. By applying $\bar{\partial}$-steepest descent method, we derive the…

Using the matrix Riemann-Hilbert factorization approach for nonlinear evolution systems which take the form of Lax-pair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, the…

In this work, the nonlinear steepest descent method is employed to study the long-time asymptotics of the integrable nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation with a step-like initial data: $q_{0}(x)\rightarrow0$ as…

We consider the Cauchy problem for the focusing nonlinear Schr\"odinger equation with initial data approaching different plane waves $A_j\mathrm{e}^{\mathrm{i}\phi_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as $x\to\pm\infty$. The goal is to…

We consider the focusing integrable nonlocal nonlinear Schr\"odinger 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…

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,…

In this paper, we mainly focus on the Cauchy problem of an integrable nonlocal Hirota equation with initial value in weighted Sobolev space. Through the spectral analysis of Lax pairs, we successfully transform the Cauchy problem of the…

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 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)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0,…

In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation (nonlocal mKdV) \[q_t(x,t)+q_{xxx}(x,t)-6q(x,t)q(-x,-t)q_x(x,t)=0,\] which can be viewed as a generalization of the…

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as $t \to \pm \infty$ $(x/t \sim {\cal O}(1))$ of the solution…

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.…

This is the second part of a two-paper series studying the nonlinear Schr\"odinger equation with quasi-periodic initial data. In this paper, we focus on the quasi-periodic Cauchy problem for the derivative nonlinear Schr\"odinger equation.…

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann-Hilbert…

In this paper, we investigate the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation $$m_{t}=b u_{x}+\frac{1}{2}\left[m\left(|u|^{2}-\left|u_{x}\right|^{2}\right)\right]_{x}-\frac{1}{2} m\left(u \bar{u}_{x}-u_{x}…

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<x<+\infty,\\ &u(x,0)=u_0(x),&&&&&&-\infty<x<+\infty,…