arXiv subjects

Gaozhan Li

Publications and source records attributed to Gaozhan Li.

4 recordsLinked to original sources

arXiv2026

Painlevé XXXIV asymptotics for the defocusing nonlinear Schrödinger equation with a finite-genus algebro-geometric background

In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr$\ddot{\text{o}}$dinger equation with a finite genus algebro-geometric background. Long-time asymptotics of the solution are derived in four space-time regions. It comes out that the leading-order term in the expansion is, up to a constant, given by the background solution with a shift of the parameter. The subleading term, however, decays at different rates for different regions. We particularly highlight that in the two transition regions, they are of order $\mathcal{O}(t^{-1/3})$ and the coefficients involve an integral of the Painlevé XXXIV transcendent. We establish our results by applying a nonlinear steepest descent analysis to the associated Riemann-Hilbert problems.

math.AP↗

arXiv2025

Riemann-Hilbert approach to the Algebro-Geometric solution of the modified Camassa-Holm equation with linear dispersion term

This paper aims at providing an exact algebro-geometric solution of the modified Camassa-Holm (mCH) equation derived from hyperelliptic curves in $4(p+q)-1$ genus. To achieve this goal, we construct the Riemann-Hilbert problems cosponsoring to the mCH equation, which can be solved exactly by the Baker-Akhiezer function. Then the precise expression of the algebro-geometric solution of the mCH equation can be obtained through reconstructed formula.

math-ph↗

arXiv2023

Long time asymptotic behavior for the nonlocal nonlinear Schrödinger equation with weighted Sobolev initial data

In this paper, we extend $\overline\partial$ steepest descent method to study the Cauchy problem for the nonlocal nonlinear Schrödinger (NNLS) equation with weighted Sobolev initial data %and finite density initial data \begin{align*} &iq_{t}+q_{xx}+2σq^2(x,t)\overline{q}(-x,t)=0, & q(x,0)=q_0(x), \end{align*} where $ q_0(x)\in L^{1,1}(\mathbb{R})\cap L^{2,1/2}(\mathbb{R})$. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem is expressed in terms of solutions of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, we obtain the asymptotic expansion of the Cauchy problem for the NNLS equation in solitonic region. The leading order term is soliton solutions, the second term is the error term is the interaction between solitons and dispersion, the error term comes from the corresponding $\bar{\partial}$ equation. Compared to the asymptotic results on the classical NLS equation, the major difference is the second and third terms in asymptotic expansion for the NNLS equation were affected by a function $ {\rm Im}ν(ξ)$ for the stationary phase point $ξ$.

math.AP↗

arXiv2022

On the long-time asymptotics of the modified Camassa-Holm equation with step-like initial data

We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with step-like initial data \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}=0, \quad m=u-u_{xx}, \nonumber \\ &u(x,0)=u_0(x)\to \left\{ \begin{array}{ll} A_1, &\ x\to+\infty,\\[5pt] A_2, &\ x\to-\infty, \end{array}\right.\nonumber \end{align} where $A_1$ and $A_2$ are two positive constants. Our main technical tool is the representation of the Cauchy problem with an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the step-like initial problem is characterized via the solution of a RH problem in the new scale $(y,t)$. We adopt double coordinates $(ξ, c)$ to divide the half-plane $\{ (ξ,c): ξ\in \mathbb{R}, \ c> 0, \ ξ=y/t\}$ into four asymptotic regions. Further using the Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution $u(y,t)$ in different space-time regions by the different choice of g-function. The corresponding leading asymptotic approximations are given with the slow/fast decay step-like background wave in genus-0 regions and elliptic waves in genus-2 regions. The second term of the asymptotics is characterized by Airy function or parabolic cylinder model. Their residual error order is $\mathcal{O}(t^{-1})$ or $\mathcal{O}(t^{-2})$ respectively.

nlin.SI↗