具有加权Sobolev初值的复短脉冲方程的孤子解
Soliton resolution for the complex short pulse equation with weighted Sobolev initial data
摘要：
在加权Sobolev空间$H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})\}$中，利用$\bar{\partial}$ -最速下降法研究了复短脉冲（CSP）方程的初值问题.在一个固定的时空锥$S(x_{1},x_{2},v_{1},v_{2})=\{(x,t)\in\mathbb{R}^{2}: y=y_{0}+vt, ~y_{0}\in[y_{1},y_{2}], ~v\in[v_{1},v_{2}]\}$中导出了解$u(x,t)$的长时间渐近性态。基于所得的渐近行为，我们证明了CSP方程的解分辨猜想，其中包括离散谱上由$N(I)$ -孤立子证实的孤立子项和连续谱上的$t^{-\frac{1}{2}}$阶项，残余误差高达$O(t^{-1})$。
英文摘要：
We employ the $\bar{\partial}$-steepest descent method in order to investigate the Cauchy problem of the complex short pulse (CSP) equation with initial conditions in weighted Sobolev space $H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})\}$. The long time asymptotic behavior of the solution $u(x,t)$ is derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})=\{(x,t)\in\mathbb{R}^{2}: y=y_{0}+vt, ~y_{0}\in[y_{1},y_{2}], ~v\in[v_{1},v_{2}]\}$. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the CSP equation which includes the soliton term confirmed by $N(I)$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-1})$.
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