论文标题
用于过渡区域中散焦的海洛塔方程的pachleve型渐近造型
Painleve-type asymptotics for the defocusing Hirota equation in transition region
论文作者
论文摘要
我们考虑了线条上经典的海洛塔方程的库奇问题，并具有衰减的初始数据。基于Hirota方程的LAX对的光谱分析，我们首先就Riemann-Hilbert问题的解决方案表示了Cauchy问题的解决方案。此外，我们应用非线性陡峭的下降分析，以获得临界过渡区域$ | \ frac {x} {x} {t} - \ frac {α^2} {3β} | t^{2/3} {2/3} \ leq m $，$ m $是一个正常常数的长期渐近分析。我们的结果表明，可以用painlevé$ \ mathrm {ii} $方程的解决方案来表达Hirota方程的长时间渐近学。关键字：海洛塔方程，最陡峭的下降方法，painlevé$ \ mathrm {ii} $方程，长期渐近学。
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.
