在本文中,我们研究初值为 q0x 的如下一维线性薛定谔方程的柯西问题：iqt+qxx=0,其中 q0(x)∈H2m,2m(R)。利用对非线性梯度共轭法,我们计算得到了与初值正则性对应的线性薛定谔方程高阶渐近展开式。
In this paper we compute the higher order long time asymptotics of the linear Schrödinger equation using the ∂̅-nonlinear steepest descent method. We assume initial condition in weighted Sobolev space with finite order of regularity and decay.
[1] Deift P, Zhou X.A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation[J]. The Annals of Mathematics, 1993, 137(2): 295. DOI: 10.2307/2946540.
[2] Fokas A S, Zakharov V E.Important Developments in Soliton Theory[M]. Cham: Springer Berlin Heidelberg, 1993. DOI: 10.1007/978-3-642-58045-1.
[3] McLaughlin K T -R, Miller P D. The $\bar{δ}$--steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights[J]. International Mathematics Research Papers, 2006: 1-78. DOI: 10.1155/imrp/2006/48673.
[4] McLaughlin K T -R, Miller P D. The $\bar{∂}$-steepest descent method for orthogonal polynomials on the real line with varying weights[J]. International Mathematics Research Notices, 2008, 2008: 1-66. DOI: 10.1093/imrn/rnn075.
[5] Dieng, M, McLaughlin K, Miller P D. Dispersive asymptotics for linear and integrable equations by the $\bar{∂}$-steepest descent method[M]. Berlin: Springer Berlin Heidelberg, 2019. DOI: 10.1007/978-1-4939-9806-7.
[6] Deift P A, Zhou X.Long-time asymptotics for integrable systems. Higher order theory[J]. Communications in Mathematical Physics, 1994, 165(1): 175-191. DOI: 10.1007/BF02099741.
