具有有限密度初值的Wadati-Konno-Ichikawa方程的长时间渐近性
Long time asymptotic for the Wadati-Konno-Ichikawa equation with finite density initial data
摘要：
本文研究了具有有限密度初值的Wadati-Konno-Ichikawa（WKI）方程的Cauchy问题.利用Deift-Zhou非线性最速下降法的$\bar{\partial}$推广，得到了时空孤子区域中解$q(x,t)$的长时间渐近性态.在此基础上，用离散谱上的$N(I)$ -孤子项和连续谱上的$t^{-\frac{1}{2}}$首阶项表征了WKI方程的渐近逼近，其残差可达$O(t^{-\frac{3}{4}})$ .我们的结果也证实了有限密度初始条件下WKI方程的孤子分辨猜想。
英文摘要：
In this work, we investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with finite density initial data. Employing the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the long time asymptotic behavior of the solution $q(x,t)$ in space-time soliton region. Based on the resulting asymptotic behavior, the asymptotic approximation of the WKI equation is characterized with the soliton term confirmed by $N(I)$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ leading order term on continuous spectrum with residual error up to $O(t^{-\frac{3}{4}})$. Our results also confirm the soliton resolution conjecture for the WKI equation with finite density initial data.
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