arXiv subjects

Qiaoyuan Cheng

Publications and source records attributed to Qiaoyuan Cheng.

6 recordsLinked to original sources

arXiv2025

Existence of global solutions to the Fokas-Lenells equation with arbitrary spectral singularities

We establish the global existence of solutions to the Fokas-Lenells equation for any initial data in a weighted Sobolev space $H^{3}(\mathbb{R})\cap H^{2,1}(\mathbb{R})$.This result removes all spectral restrictions on the initial data required in our previous work. The proof primarily relies on the inverse scattering transform formulated as new Riemann-Hilbert problems and Zhou's $L^{2}$-Sobolev bijectivity theory.

math.AP↗

arXiv2022

The Fokas-Lenells equation on the line: Global well-posedness with solitons

In this paper, we prove the existence of global solutions in $H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$ to the Fokas-Lenells (FL) equation on the line when the initial data includes solitons.A key tool in proving this result is a newly modified Darboux transformation, which adds or subtracts a soliton with given spectral and scattering parameters. In this way the inverse scattering transform technique is then applied to establish the global well-posedness of initial value problem with a finite number of solitons based on our previous results on the global well-posedness of the FL equation.

math.AP↗

arXiv2022

A Riemann-Hilbert approach to existence of global solutions to the Fokas-Lenells equation on the line

We obtain the the existence of global solutions to the Cauchy problem of the Fokas-Lenells (FL) equation on the line \begin{align} &u_{xt}+αβ^2u-2iαβu_x-αu_{xx}-iαβ^2|u|^2u_x=0,\nonumber \\ &u(x,t=0)=u_0(x), \nonumber \end{align} where without the small-norm assumption on initial data $u_0(x)\in H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$. Our main technical tool is the inverse scattering transform method based on the representation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem. The existence and the uniqueness of the RH problem is shown via a general vanishing lemma. The spectral problem associated with the FL equation is changed into an equivalent Zakharov-Shabat-type spectral problem to establish the RH problems on the real axis. By representing the solutions of the RH problem via the Cauchy integral protection and the reflection coefficients, the reconstruction formula is used to obtain a unique local solution of the FL equation. Further, the eigenfunctions and the reflection coefficients are shown Lipschitz continuous with respect to initial data, which provides a priori estimate of the solution to the FL equation. Based on the local solution and the uniformly priori estimate, we construct a unique global solution in $H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$ to the FL equation.

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↗

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↗