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 引用本文: 黄虎. 无穷多海洋表面波共振的Zakharov方程[J]. 应用数学和力学, 2014, 35(10): 1143-1150.
HUANG Hu. Zakharov-Type Equations for Resonances of an Infinite Number of Ocean Surface Waves[J]. Applied Mathematics and Mechanics, 2014, 35(10): 1143-1150. doi: 10.3879/j.issn.1000-0887.2014.10.009
 Citation: HUANG Hu. Zakharov-Type Equations for Resonances of an Infinite Number of Ocean Surface Waves[J]. Applied Mathematics and Mechanics, 2014, 35(10): 1143-1150.

## 无穷多海洋表面波共振的Zakharov方程

##### doi: 10.3879/j.issn.1000-0887.2014.10.009

###### 作者简介:黄虎（1964—），男，新疆石河子人，教授，博士，博士生导师(Tel: +86-21-56332947;E-mail: hhuang@shu.edu.cn).
• 中图分类号: O353.2

## Zakharov-Type Equations for Resonances of an Infinite Number of Ocean Surface Waves

Funds: The National Natural Science Foundation of China（11172157）
• 摘要: 从基本的波之“能量、动量、作用量”守恒定律出发，遵照普适的“对称性决定相互作用”法则和“Hamilton”结构，运用Hamilton海洋表面波复正则方程、正则变换及其Poisson括号条件，并结合经典的3-4-5-波共振条件,推导出两大类“无穷多海洋表面波相互作用的共振条件”；相应地就建立了两大类“无穷多海洋表面波共振的Zakharov方程”.以此，就为最具根本性、普遍性的海洋波湍流搭建了一个必备、先行和完备的理论框架.
•  [1] Dirac P A M.The Principles of Quantum Mechanics [M]. Oxford: Oxford University Press, 1958. [2] Weinberg S.Cosmology [M]. Oxford: Oxford University Press, 2008. [3] 严波, 刘小会, 赵莉, 周林抒. 存在内共振的覆冰四分裂导线的非线性舞动[J]. 应用数学和力学, 2014,35(1): 39-49.(YAN Bo, LIU Xiao-hui, ZHAO Li, ZHOU Lin-shu. Nonlinear galloping of iced quad-bundle conductors with internal resonances[J].Applied Mathematics and Mechanics,2014,35(1): 39-49.（in Chinese）) [4] Kartashova E.Nonlinear Resonance Analysis [M]. Cambridge: Cambridge University Press, 2011. [5] Dysthe K, Krogstad H E, Müller P. Oceanic rogue waves[J].Annual Review of Fluid Mechanics,2008,40: 287-310. [6] Phillips O M. On the dynamics of unsteady gravity waves of finite amplitude—part 1: the elementary interactions[J].Journal of Fluid Mechanics,1960,9: 193-217. [7] Hasselmann K. On the non-linear energy transfer in a gravity-wave spectrum—part 1: general theory[J].Journal of Fluid Mechanics,1962,12: 481-500. [8] Zakharov V E. Stability of periodic waves of finite amplitude on the surface of a deep fluid[J].Journal of Applied Mechanics and Technical Physics,1968,9(2): 190-194. [9] Dyachenko A I, Lvov Y V. On the Hasselmann and Zakharov approaches to the kinetic equations in gravity waves[J]. Journal of Physical Oceanography,1995,25(12): 3237-3238. [10] 杨振宁. 杨振宁文集[M]. 上海: 华东师范大学出版社, 1998.(YANG Chen-ning.Chen Ning Yang’s Collection [M]. Shanghai: The East China Normal University Publishing Press, 1998.(in Chinese)) [11] Stiassnie M, Shemer L. On modifications of the Zakharov equation for surface gravity waves[J].Journal of Fluid Mechanics,1984,143: 47-67. [12] Krasitskii V P. On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves[J].Journal of Fluid Mechanics,1994,272: 1-20. [13] Zakharov V. Statistical theory of gravity and capillary waves on the surface of a finite depth fluid[J].Eur J Mech B/Fluids,1999,18(3): 327-344. [14] 黄虎. 无穷多海洋表面波相互作用的能量守恒和共振条件[J]. 应用数学和力学, 2014,35(5): 565-571.（HUANG Hu. Energy conservation and resonance conditions for interactions of an infinite number of ocean surface waves[J].Applied Mathematics and Mechanics,2014,35(5): 565-571.（in Chinese）） [15] Zakharov V E, L’vov V S, Falkovich G.Kolmogorov Spectra of Turbulence I: Wave Turbulence[M]. Berlin: Springer, 1992. [16] McGoldrick L F. Resonant interactions among capillary-gravity waves[J].Journal of Fluid Mechanics,1965,21: 305-331. [17] McLean J W. Instabilities of finite amplitude gravity waves on water of finite depth[J].Journal of Fluid Mechanics,1982,114: 331-341. [18] Nazarenko S.Wave Turbulence [M]. Berlin: Springer, 2011. [19] Newell A C, Rumpf B. Wave turbulence[J].Annual Review of Fluid Mechanics,2011,43: 59-78.

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##### 出版历程
• 收稿日期:  2014-03-24
• 修回日期:  2014-04-14
• 刊出日期:  2014-10-15

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