arXiv subjects

Luman Ju

Publications and source records attributed to Luman Ju.

3 recordsLinked to original sources

arXiv2024

A Riemann-Hilbert approach to the two-component modified Camassa-Holm equation

In this paper, we develop a Riemann-Hilbert (RH) approach to the Cauchy problem for the two-component modified Camassa-Holm (2-mCH) equation based on its Lax pair. Further via a series of deformations to the RH problem by using the $\bar{\partial}$-generalization of Deift-Zhou steepest descent method, we obtain the long-time asymptotic approximations to the solutions of the 2-mCH equation in four kinds of space-time regions. Especially we introduce a technique to unify multi-jump matrix factorizations into one form which can greatly simplify the calculation of the $\bar{\partial}$-steepest descent method.

math-ph↗

arXiv2023

Long-time asymptotic behavior of the Hunter-Saxton equation

With $\bar{\partial}$-generalization of the Deift-Zhou steepest descent method, we investigate the long-time asymptotics of the solution to the Cauchy problem for the Hunter-Saxton (HS) equation \begin{eqnarray} &&u_{txx}-2\omega u_x+2u_xu_{xx}+uu_{xxx}=0,\quad x\in \mathbb{R},\ t>0,\nonumber\\ &&u(x,0)=u_0(x), \nonumber \end{eqnarray} where $u_0\in H^{3,4}(\mathbb{R})$ and $\omega>0$ is a constant. Using the new scale $(y,t)$ and a series of deformations to a Riemann-Hilbert problem associated with the Cauchy problem, we obtain the long-time asymptotic approximations of the solution $u(x,t)$ in two space-time regions: The solution of the HS equation decays as the speed of $\mathcal{O}(t^{-1/2})$ in the region $y/t >0$; While in the region $y/t<0$, the solution of the HS equation is depicted by a parabolic cylinder model with an residual error order $\mathcal{O}(t^{-1+\frac{1}{2p}})$ with $ p>2$.

math.AP↗

arXiv2022

Painleve-type asymptotics for the defocusing Hirota equation in transition region

We consider the Cauchy problem for the classical Hirota equation on the line with decaying initial data. Based on the spectral analysis of the Lax pair of the Hirota equation, we first expressed the solution of the Cauchy problem in terms of the solution of a Riemann-Hilbert problem. Further we apply nonlinear steepest descent analysis to obtain the long-time asymptotics of the solution in the critical transition region $|\frac{x}{t} - \frac{α^2}{3β}|t^{2/3} \leq M$, $M$ is a positive constant. Our result shows that the long time asymptotics of the Hirota equation can be expressed in terms of the solution of Painlevé $\mathrm{II}$ equation. Keywords: Hirota equation, steepest descent method, Painlevé $\mathrm{II}$ equation, long-time asymptotics.

math-ph↗