非线性双曲型 Schrödinger 方程柯西问题的解析光滑效应

数学物理学报 ›› 2025, Vol. 45 ›› Issue (5): 1463-1476.

非线性双曲型 Schrödinger 方程柯西问题的解析光滑效应

郭留涛(),徐超江^*()

南京航空航天大学数学学院南京 211100

收稿日期:2024-07-10 修回日期:2025-07-18 出版日期:2025-10-26 发布日期:2025-10-14
通讯作者: * 徐超江, E-mail:xuchaojiang@nuaa.edu.cn
作者简介:郭留涛, E-mail:guoliutao@nuaa.edu.cn
基金资助:
国家自然科学基金(12031006);中央高校基本科研业务费

Analytical Smoothing Effect on Cauchy Problem of Nonlinear Hyperbolic Schrödinger Equation

Liutao Guo(),Chaojiang Xu^*()

School of mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211100

Received:2024-07-10 Revised:2025-07-18 Online:2025-10-26 Published:2025-10-14
Supported by:
NSFC(12031006);Fundamental Research Funds for the Central Universities of China

摘要/Abstract

摘要：

该文研究一类非线性双曲型 Schrödinger 方程的 Cauchy 问题的解析光滑效应, 对于在有限阶 Sobolev 空间给定的指数衰减的 Cauchy 初始值, 我们证明了其解关于时间变量和空间变量当 $t\neq0$ 时都是解析的, 因此双曲型 Schrödinger 方程具有类似于经典 Schrödinger方程的解析光滑效应特性.

关键词: Cauchy 问题, 拟微分算子, 解析性光滑效应, 双曲型 Schrödinger 方程

Abstract:

In this paper, we study the analytical smoothing effect of Cauchy problem for a class of nonlinear hyperbolic Schrödinger equation. For the Cauchy initial value of exponential decay given in a finite Sobolev space, we prove that the solution of the equation is analytic with respect to both time and space variables when $t\neq0$. Therefore, the hyperbolic Schrödinger equation has analytical smoothing properties similar to the classical Schrödinger equation.

Key words: Cauchy problem, Quasi-differential operator, analytical smoothing effect, hyperbolic Schrödinger equation

中图分类号:

O175.2

郭留涛, 徐超江. 非线性双曲型 Schrödinger 方程柯西问题的解析光滑效应[J]. 数学物理学报, 2025, 45(5): 1463-1476.

Liutao Guo, Chaojiang Xu. Analytical Smoothing Effect on Cauchy Problem of Nonlinear Hyperbolic Schrödinger Equation[J]. Acta mathematica scientia,Series A, 2025, 45(5): 1463-1476.

参考文献

[1]	Liu C S, Tripathi V K. Laser guiding in an axially nonuniform plasma channel. Phys Plasmas, 1994, 1: 3100-3103
[2]	Gill T S. Optical guiding of laser beam in nonuniform plasma. Pramana J Phys, 2000, 55(5/6): 835-842
[3]	Bergé L. Wave collapse in physics: principles and applications to light and plasma waves. Physics Reports, 1998, 303: 259-370
[4]	Hayashi N, Saitoh S. Analyticity and smoothing effect for the Sch?dinger equation. Ann Inst Henri Poincaré, Physique Théorique, 1990, 52: 163-173
[5]	Hayashi N, Saitoh S. Analyticity and global existence of small solutions to some nonlinear Schr?dinger equations. Commun Math Phys, 1990, 129: 27-41
[6]	Hayashi N, Kato K. Analyticity in time and smoothing effect of solutions to nonlinear Sch?dinger equations. Commun Math Phys, 1997, 184: 273-300
[7]	Ghidaglia J M, Saut J C. On the initial value problem for the Davey-Stewartson systems. Nonlinearity, 1990, 3: 475-506
[8]	Klein C, McLaughlin K, Stoilov N. Spectral approach to the scattering map for the semi-classical defocusing Davey-Stewartson II equation. Phys D, 2019, 400: Art 132126
[9]	Richards G. Mass concentration for the Davey-Stewartson system. Differential Integral Equations, 2011, 24: 261-280
[10]	Forcella L. On finite time blow-up for a 3D Davey-Stewartson system. Proc Amer Math Soc, 2022, 150: 5421-5432
[11]	Feng B H, Ren J J, Wang K. Blow-up in several points for the Davey-Stewartson system in $\Bbb R^2$. J Math Anal Appl, 2018, 466: 1317-1326
[12]	Besse C, Mauser N J, Stimming H P. Numerical study of the Davey-Stewartson system. ESAIM: Mathematical Modelling and Numerical Analysis, 2004, 38(6): 1035-1054
[13]	Zhao C, Li Y, Zhou S. Asymptotic smoothing effect of solutions to Davey-Stewartson systems on the whole plane. Acta Math Sin (Engl Ser), Vol 23, 2007: 2043-2060
[14]	Li Y, Guo B. Existence and Decay of Weak Solutions to Degenerate Davey-Stewartson Equations. Acta Math Scientia, 2002, 22(3): 302-310
[15]	Hayashi N, Naumkin P I, Uchida H. Analytic smoothing effects of global small solutions to the elliptic-hyperbolic Davey-Stewartson system. Adv Differential Equations, 2002, 7: 469-492
[16]	曹晓东, 郭留涛. 一类非椭圆薛定谔方程的 Cauchy 问题的光滑性效应. 数学杂志, 2022, 42: 523-532
[16]	Cao X D, Guo L T. Smoothing effect of the Cauchy problem for a class of non-elliptic Schr?dinger equation. J Math, 2022, 42: 523-532
[17]	Guo L T, Xu C J. Free transport equation and hyperbolic Schr?dinger equation via Wigner transformation. preprint.
[18]	Shubin M. Pseudodifferential Operators and Spectral Theory. Berlin: Springer-Verlag, 1987