一类完整Coriolis力作用下的高阶非线性Schrödinger方程的推导

数学物理学报 ›› 2018, Vol. 38 ›› Issue (5): 883-892.

一类完整Coriolis力作用下的高阶非线性Schrödinger方程的推导

王丹妮,杨红丽*(),杨联贵

内蒙古大学数学科学学院呼和浩特 010021

收稿日期:2016-07-13 出版日期:2018-10-26 发布日期:2018-11-09
通讯作者: 杨红丽 E-mail:hongliyang3@sohu.com
基金资助:
国家自然科学基金(11762011)

Derivation of a Higher Order Nonlinear Schrödinger Equation with Complete Coriolis Force

Danni Wang,Hongli Yang*(),Liangui Yang

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021

Received:2016-07-13 Online:2018-10-26 Published:2018-11-09
Contact: Hongli Yang E-mail:hongliyang3@sohu.com
Supported by:
the NSFC(11762011)

摘要/Abstract

摘要：

在正压流体中，从含有完整Coriolis力的准地转位涡方程出发，采用摄动展开的方法推导了一类新的高阶非线性Schrödinger方程，用于描述地球流体力学中的非线性调制Rossby波.从方程中，讨论了调制波列.结果表明，完整Coriolis力下的水平分量和地形会影响均匀Rossby波调制不稳定，并且不稳定区域也会随着改变.此外，均匀基本流也是影响Rossby孤立波调制不稳定性的的重要因素.

关键词: 完整Coriolis力, Rossby波, 高阶非线性Schrödinger方程, 调制不稳定性

Abstract:

In this paper, based on the barotropic potential vorticity equation with complete Coriolis force, a new higher order nonlinear Schrödinger equation is derived by using perturbation expansion method to describe nonlinear modulated Rossby waves in the geophysical fluid. From this equation, the modulational wave trains are discussed. It is found that the horizontal component term of Coriolis force and the topography term has an effect on the uniform Rossby wave trains and the instability region have changed. In addition, the uniform background basic flow does affect the modulational instability of solitary Rossby wave train.

Key words: Complete Coriolis force, Rossby waves, Higher order nonlinear Schrödinger equation, Modulational disturbances

中图分类号:

P315

王丹妮,杨红丽,杨联贵. 一类完整Coriolis力作用下的高阶非线性Schrödinger方程的推导[J]. 数学物理学报, 2018, 38(5): 883-892.

Danni Wang,Hongli Yang,Liangui Yang. Derivation of a Higher Order Nonlinear Schrödinger Equation with Complete Coriolis Force[J]. Acta mathematica scientia,Series A, 2018, 38(5): 883-892.

参考文献

1	Long R R , Andrushkiw R I , Huang X H . Solitary waves in the Westerlies. J Atmos Sci, 1964, 21: 197- 200
2	Benney D J . Long nonlinear waves in fluid flows. J Math Phys, 1966, 45: 52- 63
3	Redekopp L G . On the theory of solitary Rossby waves. J Fluid Meth, 1977, 82: 725- 745
4	Boyd J P . Equatorial solitary waves. Part Ⅰ:Rossby solitons. J Phys Ocean, 1980, 10 (11): 1699- 1718
5	Grimshaw R H J . Evolution equations for long, nonlinear internal waves in stratified shear flows. Stud Appl Math, 1981, 65: 159- 188
6	Yang H W , Yin B S , Yang D I , Xu Z H . Forced solitary Rossby waves under the influence of slowly varying topography with time. Chin Phys B, 2011, 20 (12): 120201
7	Benney D J . Large amplitude Rossby waves. Stud Appl Math, 1979, 60: 1- 10
8	Yamagata T . The stability, modulation and long wave resonance of a planetary wave in a rotating, two-layer fluid on a channel beta-plane. J Meteorol Soc Japan, 1980, 58: 160- 171
9	Bao W , Jaksch D , Markowich P . Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation. J Comput Phys, 2003, 187 (1): 318- 342
10	Bao W , Cai Y . Mathematical theory and numerical methods for Bose-Einstein condensation Kinet. Relat Mod, 2013, 6: 1- 135
11	Liang X , Khaliq A , Sheng Q . Exponential time differencing Crank-Nicolson method with a quartic spline approdinger for nonlinear Schr?dinger equations. Appl Math Comput, 2014, 235: 235- 252
12	Sheng Q , Khaliq A , Al-Said E . Solving the generalized nonlinear Schr?ginger equation via quartic spline approximation. J Comput Phys, 2001, 166 (2): 400- 417
13	Peng C Q , Ma S C . The existence of nontrivial solutions for a class of asymtotically linear equation. Acta Mathematica Scientia, 2013, 33A (6): 1035- 1044
14	Xu N , Ma S W . Ground state sulutions for periodic Schr?dinger equation with critical sobolev exponent. Acta Mathematica Scientia, 2015, 35A (4): 651- 655
15	Wei G M , Li Q . Mountain pass solutions for fractional coupled nonlinear Schr?dinger systems. Acta Mathematica Scientia, 2016, 36A (1): 65- 79
16	Cai W , Wang Y , Song Y . Numerical simulation of Rogue waves by the local discontinuous galerkin method. Chin Phys Lett, 2014, 31 (4): 040201
17	Liao C C , Cui J C , Liang J Z , Ding X H . Muilti-symplectic variational integrators for nonlinear Schr?dinger equations with variable coefficients. Chin Phy B, 2016, 25 (1): 010205
18	Chen J C , Li B , Chen Y . Novel exact solutions of coupled nonlinear Schr?dinger equations with time-space modulation. Chin Phy B, 2013, 11: 110306
19	Yao Z A . Homogenization of some linear and semilinear Schr?dinger equations with real potential. Acta Mathematica Scientia, 2001, 21B (1): 137- 144
20	Li J W , Fang N W , Zhang J , et al. (2+1)-dimensional dissipation nonlinear Schr?dinger equation for envelope Rossby solitary waves and chirp effect. Chin Phy B, 2016, 25 (4): 040202
21	Luo D H . Derivation of a higher order nonlinear Schr?dinger equation for weakly nonlinear Rossby waves. Wave Motion, 2001, 33: 339- 347
22	Li Z D , Wu X , Li Q Y , He P B . Kuznetsov-Masoltion and Akhmediev breather of high-order nonlinear Schr?dinger equation. Chin Phy B, 2016, 25 (1): 010507
23	Philips N A . The equations of motion for a shallow rotating atmosphere and 'traditional approximation'. J Atmos Sci, 1966, 23 (5): 626- 628
24	Veronis G . Comments on Phillips' (1966) proposed simplification of the equations of motion for shallow rotating atmosphere. J Atmos Sci, 1968, 25 (6): 1154- 1155
25	Wangness R K . Comment on "The equations of motion for a shallow rotating atmosphere and the 'traditional approxmation'". J Atmos Sci, 1970, 27 (3): 504- 506
26	Leibovich S , Lele S K . The influence of the horizonal component of the Erath's angular velocity on the instability of the Ekman layer. Journal of Fluid Mechanics, 1985, 150: 41- 87
27	Draghici I . Non-hydrostatic Coriolis effects in an isentropic coordinate frame. Russian Meteorology and Hydrology, 1987, 17: 45- 54
28	Sun W Y . Unsymmetrical symmetric instability. Quarterly Journal of the Royal Meteorological Society, 1995, 121: 419- 431
29	White A A , Bromely R A . Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force. Quarterly Journal of the Royal Meteorological Society, 1995, 121: 399- 418
30	Burger A P . The potential vorticity equation:from planetary to small scale. Tellus, 1991, 43A: 191- 197
31	赵强, 于鑫. 完整Coriolis力作用下非线性Rossby波的精确解. 地球物理学报, 2008, 51 (5): 1304- 1308
31	Zhao Q , Yu X . Exact solutions to the nonlinear Rossby waves with a complete representation of the Coriolis force. Chinese Journal of Geophysics, 2008, 51 (5): 1304- 1308
32	宋健, 刘全生, 杨联贵. 缓变地形下β效应的Rossby代数孤立波. 地球物理学进展, 2013, 28 (4): 1684- 1688
32	Song J , Liu Q S , Yang L G . Algebraic solitary Rossby waves excited slowly changing topography and beta effect. Progress in Geophysics, 2013, 28 (4): 1684- 1688
33	Dellar P J , Salomon R . Shallow water equations with a copplete Coriolis force and topography. J Fluid Mech, 2005, 17: 106601
34	Plumb R A . The stability of small amplitude Rossby waves in a channel. Stud Appl Math, 1977, 80 (4): 705- 720