海洋表面波约化Hamilton方程的新发展:从小幅波到有限幅波的推广  

A New Development of Reduced Hamiltonian Equations for Ocean Surface Waves:an Extension From Small to Finite Amplitude

作　　者：王兆玲[1,2] 肖衡[2,3] 

机构地区：[1]潍坊学院数学与信息科学学院,山东潍坊261061 [2]省部共建高品质特殊钢冶金与制备国家重点实验室(上海大学),上海200444 [3]上海大学上海市应用数学和力学研究所,上海200072

出　　处：《应用数学和力学》2015年第11期1135-1144,共10页Applied Mathematics and Mechanics

基　　金：国家教委211工程科研启动基金(A.15-B002-09-032);国家自然科学基金(11372172)~~

摘　　要：海洋表面波的3-波至5-波约化Hamilton方程由于其对称多项式简化结构以及保能量等独特优点,得到广泛应用.但是,据相关近似假设,其适用范围局限于波陡很小的弱非线性波.于是进一步探讨下述推广问题:对一定范围内的有限幅非线性波,在足够精确意义上是否也能获得具对称多项式简化结构的约化Hamilton方程?由于涉及复杂非线性强耦合,在该重要方面至今尚未取得进展.提出基于Chebyshev(切比雪夫)多项式逼近处理精确水波方程强非线性耦合的新简化途径,导出具对称多项式简化结构的新约化Hamilton方程.新结果将波数与波陡之积为小量的弱非线性情形拓广到该积直至1.035的非线性情形.分析表明,在该范围内新结果的误差不超过5%,特别,当前述积邻近于0.9时新结果给出精确结果.The reduced 3-wave and 4-wave Hamiltoulan equations for ocean surface waves were widely used for the simplified structure with symmetric polynomial kernels and for the conser- vation of energy, etc. However, according to the related assumption for approximation in deri- vation, the range of applicability was limited to weakly nonlinear waves of small amplitude. Here the following issue was further studied： for nonlinear waves of finite amplitude within a certain range, was it also possible to obtain reduced Hamiltonian equations with symmetric pol- ynomial kernels in a sense of sufficient accuracy？ Because of complicated strongly nonlinear coupling, few development in this significant respect had been made as yet. A new approach was proposed based on the Chebyshev polynomials to best approximate the primitive water wave equations in the exact sense of strongly nonlinear coupling and derive new reduced Hamil- tonian equations with symmetric polynomial kernels. The new results exhibit an extension from a weakly nonlinear case in which the product of the wave number and the wave steepness is small to a nonlinear case in which this product goes up to about 1.035. Moreover, within this range, the approximation errors are lower than 5%, and in particular, the new results prove exact whenever the said product lies close to 0.9.

关 键 词：海洋表面波 有限波幅 新简化途径 新约化Hamilton方程 

分 类 号：O302[理学—力学]