论文标题
具有加权Sobolev初始数据的混合Schrödinger方程的长期渐近行为
Long-time asymptotic behavior of a mixed schrödinger equation with weighted Sobolev initial data
论文作者
论文摘要
我们应用$ \ bar {\ partial} $最陡的下降方法，以获取混合schrödinger方程$$ q_t+iq_ {xx} -ia（\ vert q \ vert^2q）_x -2b^2b^2 \ vert q \ vert q \ vert q \ vert q \ vert $ $ $ $ $在h^{2,2}中的加权sobolev空间中初始数据$ q_0（x）\中的初始数据的最小规律性假设（\ mathbb {r}）$。在渐近表达式中，领先订单$ \ Mathcal {o}（t^{ - 1/2}）$来自分散零件$ q_t+q_t+iq_ {xx} $和错误订单$ \ Mathcal {o}
We apply $\bar{\partial}$ steepest descent method to obtain sharp asymptotics for a mixed schrödinger equation $$ q_t+iq_{xx}-ia (\vert q \vert^2q)_x -2b^2\vert q \vert^2q=0,$$ $$q(x,t=0)=q_0(x),$$ under essentially minimal regularity assumptions on initial data in a weighted Sobolev space $q_0(x) \in H^{2,2}(\mathbb{R})$. In the asymptotic expression, the leading order term $\mathcal{O}(t^{-1/2})$ comes from dispersive part $q_t+iq_{xx}$ and the error order $\mathcal{O}(t^{-3/4})$ from a $\overline\partial$ equation
