下载:
-
波之运动, 可贯穿所有物质时空. 大致说来, 纷繁多样的波可划分为两大类: 双曲波和色散波[1]. 后者, 即以地球上最为广阔、直观的海洋表面波为其典型代表. 现代水波动力学, 始发于20世纪60年代初表面重力波第三阶四波共振的Phillips解[2]. 波-波共振在动理学方程的建立[3]、双波列的最简计算[4]、能量共享机制更为清晰的阐明[5]、非线性共振相互作用的简化[6]、实验证实[7,8]等等方面上引发了一系列经典结果. 目前, 适用于Bose-Einstein凝聚[9]、非线性光学[10]、海洋表面波[11]和内波[12]、宇宙开端的重力波[13]等几乎所有物质波运动尺度、具有极大普适性的波湍流[14-17]的立足点, 正是集中性地表现于海洋波-波共振的动理学方程[3,11,12,18-20]上.
海洋表面波, 可以是纯粹的表面张力波或重力波, 或表面张力波-重力波[17,21,22]. 最近, 从实验上观察到在表面张力波-重力波交叠处的非局部三波列共振现象[23,24]: 既可在通常的表面张力波-重力波之间发生, 也可在以往不允许的表面重力波之间发生. 这就涉及雷达直接观测表面张力波、潜艇航行等相关的一系列海洋活动、工程相关问题, 必将对表面重力波固有的频率适用范围产生影响. 波-波共振, 可以是一重的单波列或多波列[25,26]或多重的单波列或多波列[27]. 这两种情形就指向一种最基本的复合波态: 单波列之表面张力波-重力波的波-波共振. 其若从低阶向高阶以至于第n阶演化下去, 则其所对应的共振条件是什么? 未知. 则多重多波共振条件更是无从回答. 唯有前者获得精准结果, 方可使后者有所凭依, 最终求得确切答案, 也才能弥补经典深水表面重力波的Stokes波理论[28,29]和Phillips共振解[2]在共振上遗留下的历史空白, 也才使得与普适的Hamilton函数[17,30,31]密切关联的现今动力学方程[17,32]及其系综平均的动理学方程[3,11,12,18-20]得以极大扩展和统一刻画.
目前, 存在着深水表面张力波-重力波的第二阶三波共振条件[33]和深水重力波的第三阶四波[2]、第四阶五波[34]、第五阶六波[35]的共振条件. 较之单一的重力波, 表面张力波-重力波的单波列从低阶到高阶的自共振传播无疑具有更为丰富的内涵和结构. 从中, 正所谓“道生一, 一生二, 二生三, 三生万物”[36], 可对应着从四面八方滚滚涌来的万顷波涛.
基于此, 本文按照经典水波共振的Fourier-Stieltjes变换[2], 运用在科学、工程中大行其道的摄动方法[2,37,38], 力求严格推论出深水表面张力波-重力波之单波列自共振由低阶至高阶的一般通用共振定律.
-
假定存在一个无黏、无旋、不可压缩的深水海洋表面波运动. 建立直角坐标系
$O \text- xyz$ : 原点O、水平坐标${\boldsymbol{x}} = \left( {x, y} \right)$ 位于静水面上, 竖直坐标$z$ 向上为正. 速度势函数为$\phi \left( {{\boldsymbol{x}}, z, t} \right)$ , 自由表面位移为$\zeta \left( {{\boldsymbol{x}}, t} \right)$ . 其中,$t$ 为时间. 于是, 深水海洋表面波的运动方程组可表述为$ {\nabla ^2}\phi + \frac{{{\partial ^2}\phi }}{{\partial {z^2}}} = 0\;,\;\;\; - \infty < z \leqslant \zeta ({\boldsymbol{x}},t), $ $ \frac{\partial \zeta }{\partial t}-\frac{\partial \phi }{\partial z}+\nabla \phi \cdot \nabla \zeta =0, z=\zeta ({\boldsymbol{x}},t), $ $ \frac{p}{\rho }+\frac{\partial \phi }{\partial t}+g\zeta +\frac{1}{2}\bigg[{(\nabla \phi )}^{2}+{\bigg(\frac{\partial \phi }{\partial z}\bigg)}^{2}\bigg]=0,\;z=\zeta ({\boldsymbol{x}},t), $ $ \phi \to 0\;,\;\;\;{\kern 1pt} {\kern 1pt} z \to - \infty . $ 其中,
$\nabla = \bigg(\dfrac{\partial }{{\partial x}}, \dfrac{\partial }{{\partial y}}\bigg)$ ,$\rho $ 表示水的密度,$g$ 表示重力加速度,$p$ 表示表面压强, 即$ \begin{split} & p = - T' \left\{ {\frac{{{\zeta _{xx}} \big[ {1 + {{\left( {{\zeta _y}} \right)}^2}} \big] + {\zeta _{yy}} \big[ {1 + {{\left( {{\zeta _x}} \right)}^2}} \big] - 2{\zeta _{xy}}{\zeta _x}{\zeta _y}}}{{{{\big[ {1 + {{\left( {{\zeta _x}} \right)}^2} + {{\left( {{\zeta _y}} \right)}^2}} \big]}^{\frac{3}{2}}}}}} \right\}. \end{split} $ 其中,
$T'$ 表示表面张力系数.为了更容易实验和观测, 可以Fourier-Stieltjes变换[2]表征
$\zeta $ 和满足方程(1)与(4)的$\phi $ :$ \begin{split} \zeta \left( {{\boldsymbol{x}},t} \right) = \int_{\boldsymbol{k}} {{\text{d}}B\left( {{\boldsymbol{k}},t} \right){{\text{e}}^{{\text{i}}{\boldsymbol{k}} \cdot {\boldsymbol{x}}}},} ~~~~ \phi \left( {{\boldsymbol{x}},z,t} \right) = \int_{\boldsymbol{k}} {{\text{d}}A\left( {{\boldsymbol{k}},t} \right){{e} ^{kz}}{{\text{e}}^{{\text{i}}{\boldsymbol{k}} \cdot {\boldsymbol{x}}}}} . \end{split} $ 其中,
${\text{d}}B\left( {{\boldsymbol{k}}, t} \right)$ 和${\text{d}}A\left( {{\boldsymbol{k}}, t} \right)$ 可称为Fourier-Stieltjes变换,$B\left( {{\boldsymbol{k}}, t} \right)$ 和$A\left( {{\boldsymbol{k}}, t} \right)$ 则称为Fourier系数[2], 积分区域为整个波数矢量$ {\boldsymbol{k}} $ 平面,$ k = \left| {\boldsymbol{k}} \right| $ .据此, 运动学边界条件(2)和动力学边界条件(3)可以Fourier-Stieltjes变换依次表征为
$ \begin{split} {\text{d}}A\left( {\boldsymbol{k}},t \right) =& {k^{ - 1}}{\text{d}}B'\left( {\boldsymbol{k}},t \right) - \int_{{{\boldsymbol{k}}_1}} {{D_1}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right){\text{d}}B'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_1}},t \right)} \\ & - {k^{ - 1}}\int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {{D_2}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right){\text{d}}B'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_2}},t \right)} } \\ & - {k^{ - 1}}\int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {\int_{{{\boldsymbol{k}}_3}} {{D_3}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2},{{\boldsymbol{k}}_3}} \right){\text{d}}B'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_2}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_3}},t \right)} } } , \end{split}$ $ \begin{split} & - T\left[ {{k^2}{\text{d}}B \left( {\boldsymbol{k}},t \right) + \int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {{H_5}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right){\text{d}}B\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_2}},t \right)} } } \right] \\ {\text{ = }}&{k^{ - 1}}{\text{d}}B''\left( {\boldsymbol{k}},t \right) + g{\text{d}}B\left( {\boldsymbol{k}},t \right) + \int_{{{\boldsymbol{k}}_1}} {{H_1}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right){\text{d}}B''\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_1}},t \right)} \\ & + \int_{{{\boldsymbol{k}}_1}} {{H_2}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right){\text{d}}B'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}B'\left( {{{\boldsymbol{k}}_1}},t \right)} \\ & - \int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {{H_3}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right){\text{d}}B''\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_2}},t \right)} } \\ & - \int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {{H_4}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right){\text{d}}B'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}B'\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_2}},t \right)} } \\ & - \int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {\int_{{{\boldsymbol{k}}_3}} {{H_6}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2},{{\boldsymbol{k}}_3}} \right){\text{d}}B''\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_2}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_3}},t \right)} } } \\ & - \int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {\int_{{{\boldsymbol{k}}_3}} {{H_7}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2},{{\boldsymbol{k}}_3}} \right){\text{d}}B' \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}},t \right){\text{d}}B' \left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_2}},t \right){\text{d}}B\left( {{{\boldsymbol{k}}_3}},t \right)} } } . \end{split} $ 其中,
$T = {{T'} \mathord{\left/ {\vphantom {{T'} \rho }} \right. } \rho }$ ,$ {\text{d}}A $ 和$ {\text{d}}B $ 为简洁起见而忽略了自变量$t$ ,$ B' $ 和$ B'' $ 依次代表对时间的第一、第二阶导数,${D_n}$ $\left( {n = 1, 2} \right)$ 和${H_n}$ $\left( {n = 1, 2, 3, 4} \right)$ 的详细表达式可见文献[2]. 此外,$ \begin{split} & {D_3}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2},{{\boldsymbol{k}}_3}} \right) \\ =& \left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|\left[ {\frac{{{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}^2}}}{6} + \frac{{{{\boldsymbol{k}}_1} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{2}} \right] \\ & - \frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}\left[ {\frac{1}{2}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|}^2} + {{\boldsymbol{k}}_1} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right)} \right] \\ & - \left[ {\frac{{{\boldsymbol{k}} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right|}}} \right]\left[ {\frac{1}{2}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}^2} + {{\boldsymbol{k}}_2} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)} \right. \\ & - \left. { \frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right)\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right] \text{; } \end{split} $ $ \begin{split} {H_5}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right) =& \frac{1}{2}\left[ {{{ {{(k_{0x})^2}} }}{k_{1y}}{k_{2y}} + {{ {{(k_{0y})^2}} }}{k_{1x}}{k_{2x}}} \right] + \frac{3}{2}\left[ {{{ {{(k_{0x})^2}} }}{k_{1x}}{k_{2x}} + {{ {{(k_{0y})^2}} }}{k_{1y}}{k_{2y}}} \right] \\ & + {k_{0x}}{k_{0y}}\left( {{k_{1x}}{k_{2y}} + {k_{1y}}{k_{2x}}} \right) \text{, } \end{split} $ $ \begin{split} & {H_6}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2},{{\boldsymbol{k}}_3}} \right)\\ =& \frac{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}{k}\left[ {\frac{{{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}^2}}}{6} + \frac{{{{\boldsymbol{k}}_1} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{2}} \right] \\ &- \frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{k\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}\left[ {\frac{1}{2}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|}^2} + {{\boldsymbol{k}}_1} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right)} \right] \\ & + \left[ {1 - \frac{{{\boldsymbol{k}} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right)}}{{k\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right|}}} \right]\left[ {\frac{1}{2}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}^2} + {{\boldsymbol{k}}_2} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)} \right. \\ &-\left. { \frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right)\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right] \\ &+ \frac{1}{2}\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|\frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}} - \frac{1}{6}{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|^2}, \end{split} $ $ \begin{split} & {H_7}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2},{{\boldsymbol{k}}_3}} \right)\\ = \;&\frac{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}{k}\left[ {\frac{{{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}^2}}}{2} + \frac{{\left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_2} + {{\boldsymbol{k}}_3}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{2}} \right] \\ & - \left[ {\left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) + {{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|}^2}} \right]\frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{k\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}} \\ & - \left[ {{{\boldsymbol{k}}_3} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) + \frac{1}{2}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}^2}} \right]\frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{k\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}} \\ & - \frac{{{\boldsymbol{k}} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right)}}{{k\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right|}}\left[ {\frac{1}{2}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}^2} + {{\boldsymbol{k}}_2} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)} \right. \\ &-\left. { \frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right)\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right] \\ & + \left[ {1 - \frac{{{\boldsymbol{k}} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_2}} \right)}}{{k\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_2}} \right|}}} \right]\left[ {\frac{1}{2}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}^2} + {{\boldsymbol{k}}_1} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)} \right. \\ &- \left. { \frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right)\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right] \\ & + \left[ {1 - \frac{{{\boldsymbol{k}} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_3}} \right)}}{{k\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_3}} \right|}}} \right]\left[ {\frac{1}{2}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}^2} + {{\boldsymbol{k}}_2} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)} \right. \\ &-\left. { \frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_3}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right] \\ & + \frac{1}{2}{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|^2}\frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}} \\ & + \frac{1}{2}\left[ {1 - \frac{{{{\boldsymbol{k}}_1} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right)}}{{{k_1}\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right|}}} \right]\left[ {\frac{1}{2}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}^2} + {{\boldsymbol{k}}_2} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)} \right. \\ &- \left. { \frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right)\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right] \\ & + \frac{1}{2}\left[ {1 - \frac{{\left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_2} + {{\boldsymbol{k}}_3}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_2} + {{\boldsymbol{k}}_3}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right]\left[ {\frac{{k_1^2}}{2} + {{\boldsymbol{k}}_1} \cdot {{\boldsymbol{k}}_2}} \right. \\ &- \left. { \frac{{{{\boldsymbol{k}}_1} \cdot \left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right)\left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right) \cdot \left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_2} + {{\boldsymbol{k}}_3}} \right)}}{{{k_1}\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right|}}} \right] + \frac{1}{2}\left[ {{k_1}\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right| + {{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|}^2}} \right] \\ & \times \left\{ {\left[ {1 - \frac{{{{\boldsymbol{k}}_1} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right)}}{{{k_1}\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|}}} \right]\frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right\} + \frac{1}{2}\Big[ {{{\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right|}^2}} \\ &+ \left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right| \Big] \left[ {1 - \frac{{\left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right]\frac{{{{\boldsymbol{k}}_1} \cdot \left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right)}}{{{k_1}\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right|}} \\ & - \frac{1}{2}\left\{ {\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_3}} \right|\left[ {1 - \frac{{\left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_3}} \right|}}} \right]\frac{{{{\boldsymbol{k}}_1} \cdot \left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right)}}{{{k_1}\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_3}} \right|}}} \right\} \\ & \times \left[ {\frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_3}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_3}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right] - \frac{1}{2}\left[ {\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_2}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|\frac{{{{\boldsymbol{k}}_1} \cdot \left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_2}} \right)}}{{{k_1}\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_2}} \right|}}} \right] \\ & \times \left\{ {\left[ {1 - \frac{{\left( {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right)}}{{\left| {{{\boldsymbol{k}}_1} + {{\boldsymbol{k}}_2}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|}}} \right]\frac{{\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right) \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}} \right|\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right\}\\ & - \frac{1}{2}{\left[ {{k_1} + \left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|} \right]^2}\left[ {1 - \frac{{{{\boldsymbol{k}}_1} \cdot \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right)}}{{{k_1}\left| {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}} \right|}}} \right]. \end{split} $ -
为求解方程(8), 可对Fourier-Stieltjes变换
${\text{d}}B\left( {k, t} \right)$ 进行摄动级数展开:$ {\text{d}}B\left( {{\boldsymbol{k}},t} \right) = {\text{d}}{B_1}\left( {{\boldsymbol{k}},t} \right) + {\text{d}}{B_2}\left( {{\boldsymbol{k}},t} \right) + {\text{d}}{B_3}\left( {{\boldsymbol{k}},t} \right) + {\text{d}}{B_4}\left( {{\boldsymbol{k}},t} \right) + \cdots . $ 相应地, 可得:
$ \zeta \left( {{\boldsymbol{x}},t} \right) = {\zeta _1}\left( {{\boldsymbol{x}},t} \right) + {\zeta _2}\left( {{\boldsymbol{x}},t} \right) + {\zeta _3}\left( {{\boldsymbol{x}},t} \right) + {\zeta _4}\left( {{\boldsymbol{x}},t} \right) + \cdots . $ 将(13)式代入(8)式, 可得第一阶的微分方程和第二、三、四阶的各自积分微分方程:
$ {k^{ - 1}}{\text{d}} B_1^{\prime \prime }\left( {{\boldsymbol{k}},t} \right) + \left( {g + T{k^2}} \right){\text{d}}{B_1}\left( {{\boldsymbol{k}},t} \right) = 0, $ $ \begin{split} & {k^{ - 1}}{\text{d}} B_2 ^{\prime \prime }\left( {{\boldsymbol{k}},t} \right) + \left( {g + T{k^2}} \right){\text{d}}{B_2}\left( {{\boldsymbol{k}},t} \right)\\ =& - \int_{{{\boldsymbol{k}}_1}} {{H_1}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right){\text{d}} B_1 ^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d }}{B_1}\left( {{{\boldsymbol{k}}_1},t} \right)} - \int_{{{\boldsymbol{k}}_1}} {{H_2}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right){\text{d}} B_1^\prime \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}} B_1^\prime \left( {{{\boldsymbol{k}}_1},t} \right),} \end{split} $ $ \begin{split} & {k^{ - 1}}{\text{d}} B_3^{\prime \prime }\left( {{\boldsymbol{k}},t} \right) + \left( {g + T{k^2}} \right){\text{d}}{B_3}\left( {{\boldsymbol{k}},t} \right) \\ = & - \int_{{{\boldsymbol{k}}_1}} {{H_1}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right)\left[ {{\text{d}} B_1^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}}{B_2}\left( {{{\boldsymbol{k}}_1},t} \right) + {\text{d}} B_2^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1},t} \right)} \right]} \\ & - \int_{{{\boldsymbol{k}}_1}} {{H_2}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right)\left[ {{\rm{d}} B_1^\prime \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}} B_2^\prime \left( {{{\boldsymbol{k}}_1},t} \right)} \right.\left. { + {\text{d}} B_2^\prime \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}} B_1^\prime \left( {{{\boldsymbol{k}}_1},t} \right)} \right]} \\ & + \int_{{{\boldsymbol{k}}_1}} {\int_{ {{\boldsymbol{k}}_2}} {{H_3}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right){\text{d}} B_1^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2},t} \right)} } \\ & + \int_{{{\boldsymbol{k}}_1}} {\int_{ {{\boldsymbol{k}}_2}} {{H_4}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right){\text{d}} B_1^\prime \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2},t} \right){\text{d}} B_1^\prime \left( {{{\boldsymbol{k}}_1},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2},t} \right)} } \\ & - T\int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {{H_5}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right){\text{d}}{B_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2},t} \right),} } \end{split} $ $ \begin{split} & {k^{ - 1}}{\text{d}} B_4 ^{\prime \prime }\left( {{\boldsymbol{k}},t} \right) + \left( {g + T{k^2}} \right){\text{d}}{B_4}\left( {{\boldsymbol{k}},t} \right) \\ =& - \int_{{{\boldsymbol{k}}_1}} {{H_1}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right)\left[ {{\text{d}} B_1^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}}{B_3}\left( {{{\boldsymbol{k}}_1},t} \right) + {\text{d}} B_3^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1},t} \right)} \right.} +\left. { 2{\text{d}} B_2^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}}{B_2}\left( {{{\boldsymbol{k}}_1},t} \right)} \right] \\ & - \int\nolimits_{{\boldsymbol k_1}} {{H_2}\left( {{\boldsymbol{k}}, {{\boldsymbol{k}}_1}} \right)} \left[ {{\text{d}}B_1'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}}B_3'\left( {{{\boldsymbol{k}}_1},t} \right) + {\text{d}}B_3'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}}B_1'\left( {{{\boldsymbol{k}}_1},t} \right)} \right] + 2{\text{d}}B_2'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1},t} \right){\text{d}}B_2'\left( {{{\boldsymbol{k}}_1},t} \right) \\ & +\int_{{\boldsymbol k_1}} {\int_{{\boldsymbol k_2}} {{H_3}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_{\text{1}}},{{\boldsymbol{k}}_2}} \right)} } \left[ {{\text{d}}B_1''\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2},t} \right){\text{d}}{B_2}} \right.\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right) \\ &+ \left. {{\text{ d}}B_2''\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right) + {\text{d}}B_1''\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_2}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right)} \right] \\ & +\int_{{\boldsymbol k_1}} {\int_{{\boldsymbol k_2}} {{H_4}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_{\text{1}}},{{\boldsymbol{k}}_2}} \right)} } \left[ {{\text{d}}B_1'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2},t} \right){\text{d}}B_2'} \right.\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right) \\ &+ \left. {{\text{ d}}B_2'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2},t} \right){\text{d}}{B_1'}\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right) + {\text{d}}B_1'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2},t} \right){\text{d}}B_1'\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_2}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right)} \right] \\ & - T\int_{{\boldsymbol k_1}} {\int_{{\boldsymbol k_2}} {{H_5}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_{\text{1}}},{{\boldsymbol{k}}_2}} \right)} } \left[ {{\text{d}}B_1\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2},t} \right){\text{d}}{B_2}} \right.\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right) \\ &+ \left. {{\text{ d}}B_2^{}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right) + {\text{d}}B_1^{}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_2}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right)} \right] \\ & + \int_{{\boldsymbol k_1}} {\int_{{\boldsymbol k_2}} {\int_{{\boldsymbol k_3}} {{H_6}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_{\text{1}}},{{\boldsymbol{k}}_2},{{\boldsymbol{k}}_3}} \right)} } } \left. {{\text{d}}B_1''\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3},t} \right){\text{d}}{B_1}} \right.\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{3}}},t} \right) \\ & + \int_{{\boldsymbol k_1}} {\int_{{\boldsymbol k_2}} {\int_{{\boldsymbol k_3}} {{H_7}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_{\text{1}}},{{\boldsymbol{k}}_2},{{\boldsymbol{k}}_3}} \right)} } } \left. {{\text{d}}B_1'\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_{\text{1}}} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3},t} \right){\text{d}}B_1'} \right.\left( {{{\boldsymbol{k}}_{\text{1}}},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{2}}},t} \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_{\text{3}}},t} \right). \\[-12pt] \end{split}$ -
为规范、简要、明确地表示单波在一对或多对正负波数矢量点上的哪一阶共振与否的Fourier系数
$B$ 及其表达式里出现的系数$C$ , 特做出以下上下标规定.1) 第一阶各量只有一个下标, 均为1.
2) 第二阶各量包含以一个减号相隔的两个下标: 第1个下标均为2; 第2个下标以一个奇数
$m$ (表示“非共振”)或偶数“$m + 1$ ”(表示“共振”) 表征. 某一对正负波数矢量点即以$\left( {m, m + 1} \right)$ 刻画. 第2个下标也可围绕$\left( {m, m + 1} \right)$ 而表征为“$m, m + 1$ ”.3) 第三阶各量包含以两个减号相隔的3个下标: 第1个下标均为3; 第2个下标均为1, 即代表单波唯一一对第二阶正负波数矢量点处的非共振位移; 第3个下标同样以一个奇数
$m$ (表示“非共振”)或偶数“$m + 1$ ”(表示“共振”)表征, 同样也可表征为“$m, m + 1$ ”.4) 第四阶各量包含以三个减号相隔的4个下标: 第1个下标均为4; 第2, 3个下标皆为1, 依次表示在第二、三阶正负波数矢量点上的非共振位移; 第4个下标照样以一个奇数
$m$ (表示“非共振”)或偶数“$m + 1$ ”(表示“共振”)表征, 照样亦可表征为“$m, m + 1$ ”. -
解方程(15), 可求得:
$ \begin{split} {\text{d}}{B_1}\left( {{\boldsymbol{k}},t} \right) =& \frac{1}{2}a\left[ \delta \left( {{\boldsymbol{k}} - {{\boldsymbol{K}}_1}} \right){{\text{e}}^{ - {\text{i}}{n_1}t}}\right. \\ &+\left.\delta \left( {{\boldsymbol{k}} + {{\boldsymbol{K}}_1}} \right){{\text{e}}^{{\text{i}}{n_1}t}} \right]{\text{d}}{\boldsymbol{k}}, \end{split} $ $ {\zeta _1}\left( {{\boldsymbol{x}},t} \right) = a\cos \left( {{{\boldsymbol{K}}_1} \cdot {\boldsymbol{x}} - {n_1}t} \right)\text{, } $ $ {n_1} = {\left( {g{K_1} + TK_1^3} \right)^{{1}/{2}}}. $ 其中,
${K_1} = \left| {{{\boldsymbol{K}}_1}} \right|$ ,$\delta \left( {\boldsymbol{k}} \right)$ 表示Dirac$\delta $ 函数. (21)式即为单波列第一阶色散关系. 实际上, 这正是后续高阶共振条件的内在本质体现. -
将(19)式代入(16)式, 可得对积分有非零贡献的一对正负波数矢量点
${\boldsymbol{k}}$ :$\left( { \pm 2{{\boldsymbol{K}}_1}} \right)\left( {1, 2} \right)$ . 假设:$ {B_2}\left( {{\boldsymbol{K}},t} \right) = \int_{K' - \varepsilon }^{K' + \varepsilon } {\int_{K'' - \varepsilon }^{K'' + \varepsilon } {{\text{d}}{B_2}\left( {{\boldsymbol{k}},t} \right)} } . $ 其中,
${\boldsymbol{K}} = \left( {K', K''} \right) = \pm 2{{\boldsymbol{K}}_1}$ 或${\boldsymbol{K}} = \left( {K', K''} \right) = {\bf{0}}$ ,$\varepsilon $ 为一个任意小的正数. 由(22)式可将(16)式化为$ \begin{split} & B_2 ''\left( {{\boldsymbol{K}},t} \right) +\left( {gK + T{K^3}} \right){B_2}\left( {{\boldsymbol{K}},t} \right) \\ =\;& - K\int_{\boldsymbol{k}} \int_{{{\boldsymbol{k}}_1}} \left[ {{H_1}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right){\text{d}} B_1''( {\boldsymbol{k}} } \right. - {{\boldsymbol{k}}_1},t ){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1},t} \right) \\ & + {H_2}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right){\text{d}} B_1' ( {\boldsymbol{k}} -\left. {{\boldsymbol{k}}_1},t ){\text{d}} B_1' \left( {{{\boldsymbol{k}}_1},t} \right) \right].\\[-10pt] \end{split} $ 其中, 在上述积分上下限处对
${\boldsymbol{k}}$ 求积分,$K = \left| {\boldsymbol{K}} \right|$ . 当${\boldsymbol{K}} = \pm 2{{\boldsymbol{K}}_1}$ 时,${\boldsymbol{k}} = \pm 2{{\boldsymbol{K}}_1}$ ,${{\boldsymbol{k}}_1} = \pm {{\boldsymbol{K}}_1}$ . 则由(19)式, 可将(23)式化简为$ \left[ {\frac{{{{\text{d}}^2}}}{{{\text{d}}{t^2}}} + \left( {2{K_1}g + 8K_1^3T} \right)} \right]{B_{2 - 1,2}} = {C_{2 - 1,2}}{{\text{e}}^{ \mp 2{\text{i}}{n_1}t}}. $ 其中,
$ {B_{2 - 1,2}} = {B_{2 - 1,2}}\left( { \pm 2{{\boldsymbol{K}}_1},t} \right), {C_{2 - 1,2}} = - \frac{1}{2}{a^2}n_{\text{1}}^2{K_{\text{1}}}. $ 当
$ 2{K_1}g + 8 K_1^3 T \ne {\left( { \pm 2{n_1}} \right)^2} $ 时, 可以求得(24)式的有界非共振特解:$ {B_{2 - 1}} = {C_{2 - 1}}{{\text{e}}^{ \mp 2{\text{i}}{n_1}t}} . $ 其中,
$ {C_{2 - 1}} = \frac{{{C_{2 - 1, 2}}}}{{2{K_1}g + 8 K_1^3 T - 4 n_1^2}}. $ 当
$ 2{K_1}g + 8 K_1^3 T = {\left( { \pm 2{n_1}} \right)^2} $ 时, 即可求得单波列第二阶共振条件:$ {K_1} = \sqrt {\frac{g}{{2 T}}} . $ 据此, 可求得(24)式随时间线性增长的共振特解:
$ {B_{2 - 2}} = \pm {\rm{i}}{C_{2 - 2}}t{{\text{e}}^{ \mp 2{\text{i}}{n_1}t}}. $ 其中,
$ {C_{2 - 2}} = \frac{{{C_{2 - 1,2}}}}{{4{n_1}}} . $ -
由(26)式可得:
$ \begin{split} {\text{d}}{B_{2 - 1}}\left( {{\boldsymbol{k}}, t} \right) =\;& {C_{2 - 1}}\left[ \delta \left( {{\boldsymbol{k}} - 2{{\boldsymbol{{ K}}}_1}} \right){{\text{e}}^{ - 2{\text{i}}{n_1}t}}\right. \\ &+ \left.\delta \left( {{\boldsymbol{k}} + 2{{\boldsymbol{{ K}}}_1}} \right){{\text{e}}^{2{\text{i}}{n_1}t}} \right]{\text{d}}{\boldsymbol{k}}. \end{split} $ 将(19)式和(31)式代入(17)式, 可给出有积分贡献的两对正负波数矢量点
${\boldsymbol{k}}$ :$\left( { \pm 3{{\boldsymbol{K}}_1}} \right)\left( {1, 2} \right)$ ;$\left( { \pm {{\boldsymbol{K}}_1}} \right) \left( {3, 4} \right)$ . 与前类似, 在${\boldsymbol{k}}$ 的小区域内对(17)求${\boldsymbol{k}}$ 积分, 可得:$\begin{split} & B_{3 - 1} ^{\prime \prime }\left( {\boldsymbol{K}},t \right) + \left( {gK + T{K^3}} \right){B_{3 - 1}}\left( {\boldsymbol{K}},t \right) \\ = &- K\int_{\boldsymbol{k}} {\int_{{{\boldsymbol{k}}_1}} {\left\{ {{H_1}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}}\right)\left[ {{\text{d}} B_1^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}{B_{2 - 1}}\left( {{{\boldsymbol{k}}_1}},t \right) + {\text{d}} B_{2-1}^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1}},t \right)} \right]} \right.} } \\ &+ \left. { {H_2}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right)\left[ {{\text{d}} B_1^\prime \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}} B_{2-1}^\prime \left( {{{\boldsymbol{k}}_1}},t \right) + {\text{d}} B_{2 - 1}^\prime \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}} B_1^\prime \left( {{{\boldsymbol{k}}_1}},t \right)} \right]} \right\} \\ &+ K\int_{\boldsymbol{k}} {\int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {\left[ {{H_3}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right){\text{d}} B_1^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right)} \right.} } } \\ &+ \left. { {H_4}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right){\text{d}} B_1^\prime \left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}} B_1^\prime \left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right)} \right] \\ & - KT\int_{\boldsymbol{k}} {\int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {{H_5}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right){\text{d}}{B_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right)} } } . \end{split} $ 由(32)式, 可令
${\boldsymbol{K}}$ 分别等同于上述两对波数矢量点${\boldsymbol{k}}$ , 则可得如下方程:$ \begin{split} \left[ {\frac{{\text{d}}^2}{{{\text{d}}{t^2}}} + \left( {3{K_1}g + 27K_1^3T} \right)} \right]{B_{3 - 1 - 1,2}} = {C_{3 - 1 - 1,2}}{{\text{e}}^{ \mp 3{\text{i}}{n_1}t}}, \end{split} $ $ \left[ {\frac{{\text{d}}^2}{{{\text{d}}{t^2}}} + \left( {{K_1}g + K_1^3T} \right)} \right]{B_{3 - 1 - 3,4}} = {C_{3 - 1 - 3,4}}{{\text{e}}^{ \mp {\text{i}}{n_1}t}}. $ 其中,
$ \begin{split} {C_{3 - 1 - 1,2}} = & - \frac{{3{a^3}K_1^2}}{{16}}\left[ 2n_1^2\left( {\frac{{4n_1^2}}{{g{K_{\text{1}}} - 2TK_1^3}} - 1} \right)\right. + \left. 27TK_1^3 \right], \end{split} $ $ \begin{split} {C_{3 - 1 - 3,4}} = &\frac{{{a^3}K_1^2}}{{16}}\Bigg[ - 3TK_1^3 + 4n_1^2\left( {1 + \frac{{n_1^2}}{{g{K_1} - 2TK_1^3}}} \Bigg) \right]. \end{split} $ 当
$ 3{K_1}g + 27 K_1^3 T \ne {\left( { \pm 3{n_1}} \right)^2} $ 时, (33)式的有界非共振特解为$ {B_{3 - 1 - 1}} = {C_{3 - 1 - 1}}{{\text{e}}^{ \mp 3{\text{i}}{n_1}t}} . $ 其中,
$ {C_{3 - 1 - 1}} = \frac{{{C_{31 - 1 - 1,2}}}}{{3g{K_1} + 27TK_1^3 - 9n_1^2}}. $ 当
$ 3{K_1}g + 27 K_1^3 T = {\left( { \pm 3{n_1}} \right)^2} $ 时, 即得到单波列第三阶共振条件:$ {K_1} = \sqrt {\dfrac{g}{{3 T}}}, $ 和(33)式随时间线性增长的共振特解:
$ {B_{3 - 1 - 2}} = \pm {\text{i}}{C_{3 - 1 - 2}}t{{\text{e}}^{ \mp 3{\text{i}}{n_1}t}} . $ 其中,
$ {C_{3 - 1 - 2}}{\text{ = }}\frac{{{C_{3 - 1 - 1,2}}}}{{6{n_1}}}. $ 显见, (34)式满足共振-色散关系(21)式, 则可得其随时间线性增长的共振特解:
$ {B_{3 - 1 - 4}} = \pm {\text{i}}t{C_{3 - 1 - 4}}{{\text{e}}^{ \mp {\text{i}}{n_1}t}}. $ 其中,
$ {C_{3 - 1 - 4}} = \frac{{{C_{3 - 1 - 3,4}}}}{{2{n_1}}}. $ -
由前述可判知, 若要发生第四阶共振, 则须不发生第二、三阶共振. 为此, 由(37)式可得:
$ \begin{split} {\text{d}}{B_{3 - 1 - 1}}\left( {{\boldsymbol{k}},t} \right) =& {C_{3 - 1 - 1}}\left[ \delta \left( {{\boldsymbol{k}} - 3{{\boldsymbol{{ K}}}_1}} \right){{\text{e}}^{ - 3{\text{i}}{n_1}t}}\right. \\ &+\left. \delta \left( {{\boldsymbol{k}} + 3{{\boldsymbol{{ K}}}_1}} \right){{\text{e}}^{{\text{3i}}{n_1}t}} \right]{\text{d}}{\boldsymbol{k}}. \end{split} $ 将(19)式、(31)式、(44)式代入(18)式, 可给出有积分贡献的两对正负波数矢量点
${\boldsymbol{k}}$ :$\left( { \pm 4{{\boldsymbol{K}}_1}} \right)\left( {1, 2} \right)$ ;$\left( { \pm 2{{\boldsymbol{K}}_1}} \right)\left( {3, 4} \right)$ . 再与前类似, 在${\boldsymbol{k}}$ 的小区域内对(18)式求${\boldsymbol{k}}$ 积分, 可得:$ \begin{split} & {B''_{4 - 1 - 1}}\left( {\boldsymbol{K}},t \right) + \left( {gK + T{K^3}} \right){B_{4 - 1 - 1}}\left( {\boldsymbol{K}},t \right) \\ = & - K\int_{\boldsymbol{k}} {\int_{{{\boldsymbol{k}}_1}} {\left\{ {{H_1}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right)\left[ {{\text{d}} B_1^{\prime \prime }\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}{B_{3 - 1 - 1}}\left( {{{\boldsymbol{k}}_1}},t \right) + {\text{d}}{ B''_{3 - 1 - 1}}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1}},t \right)} \right.} \right.} } \\ & + \left. { 2{\text{d}}{ B''_{2 - 1}}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}{B_{2 - 1}}\left( {{{\boldsymbol{k}}_1}},t \right)} \right] + {H_2}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1}} \right)\left[ {{\text{d}}{ B'_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}{ B'_{3 - 1 - 1}}\left( {{{\boldsymbol{k}}_1}},t \right)} \right. \\ &+\left. { \left. { {\text{d}}{B'_{3 - 1 - 1}}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}{ B'_1}\left( {{{\boldsymbol{k}}_1}},t \right) + 2{\text{d}}{ B'_{2 - 1}}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1}},t \right){\text{d}}{B'_{2 - 1}}\left( {{{\boldsymbol{k}}_1}},t \right)} \right]} \right\} \\ & + K\int_{\boldsymbol{k}} {\int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {\left\{ {{H_3}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right)\left[ {{\text{d}}{ B''_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B_{2 - 1}}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right)} \right.} \right.} } } \\ & + \left. {{\text{d}}{ B''_{2 - 1}}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right) + {\text{d}}{ B''_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_{2 - 1}}\left( {{{\boldsymbol{k}}_2}},t \right)} \right] \\ & + {H_4}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right)\left[ {{\text{d}}{ B'_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B'_{2 - 1}}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right)} \right. \\ & + \left. {{\text{d}}{B'_{2 - 1}}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B'_1}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right) + {\text{d}}{B'_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B'_1}\left( {{{\boldsymbol{k}}_1}},t \right)d{B_{2 - 1}}\left( {{{\boldsymbol{k}}_2}},t \right)} \right] \\ & - T{H_5}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2}} \right)\left[ {{\text{d}}{B_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B_{2 - 1}}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right)} \right. \\ &+\left. { \left. {{\text{d}}{B_{2 - 1}}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right) + {\text{d}}{B_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_{2 - 1}}\left( {{{\boldsymbol{k}}_2}},t \right)} \right]} \right\} \\ & + K\int_{\boldsymbol{k}} {\int_{{{\boldsymbol{k}}_1}} {\int_{{{\boldsymbol{k}}_2}} {\int_{{{\boldsymbol{k}}_3}} {\left[ {{H_6}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2},{{\boldsymbol{k}}_3}} \right){\text{d}}{B''_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_3}},t \right)} \right.} } } } \\ &+\left. { {H_7}\left( {{\boldsymbol{k}},{{\boldsymbol{k}}_1},{{\boldsymbol{k}}_2},{{\boldsymbol{k}}_3}} \right){\text{d}}{B'_1}\left( {{\boldsymbol{k}} - {{\boldsymbol{k}}_1} - {{\boldsymbol{k}}_2} - {{\boldsymbol{k}}_3}},t \right){\text{d}}{B'_1}\left( {{{\boldsymbol{k}}_1}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_2}},t \right){\text{d}}{B_1}\left( {{{\boldsymbol{k}}_3}},t \right)} \right]. \end{split} $ 进而, 在(45)式中可令
${\boldsymbol{K}} = {\boldsymbol{k}} = \pm 4{{\boldsymbol{K}}_1}$ , 则得到:$\left[ {\frac{{\text{d}}^2}{{{\text{d}}{t^2}}} + 4\left( {g{K_1} + 16TK_1^3} \right)} \right]{B_{4 - 1 - 1 - 1,2}} = {C_{4 - 1 - 1 - 1,2}}{{\text{e}}^{ \mp 4{\text{i}}{n_1}t}}. $ 其中,
$ \begin{split} {C_{4 - 1 - 1 - 1,2}} =& - \frac{{{a^4}{K_1}n_1^2}}{4}\Bigg\{ {\frac{3}{{2\left( {g - 3TK_1^2} \right)}}} \left[ {2n_1^2\left( {\frac{{4n_1^2}}{{g - 2TK_1^2}} - {K_1}} \right) + 27TK_1^4} \right] \\ &+ \frac{{n_1^2}}{{g - 2TK_1^2}}\left( {\frac{{{\text{4}}n_1^2}}{{g - 2TK_1^2}} - 6{K_1} + \frac{{{\text{120}}TK_1^4}}{{n_1^2}}} \right) + {K_1^2} \Bigg\}. \end{split} $ 当
$ 4{K_1}g + 64 K_1^3 T \ne {\left( { \pm 4{n_1}} \right)^2} $ 时, 可得(46)式的有界非共振解:$ {B_{4 - 1 - 1 - 1}} = {C_{4 - 1 - 1 - 1}}{{\text{e}}^{ \mp 4{\text{i}}{n_1}t}}.$ 其中,
$ {C_{4 - 1 - 1 - 1}} = \frac{{{C_{4 - 1 - 1 - 1,2}}}}{{4\left( {g{K_1} + 16TK_1^3} \right) - 16n_1^2}}. $ 当
$ 4{K_1}g + 64 K_1^3 T = {\left( { \pm 4{n_1}} \right)^2} $ 时, 即得到单波列第四阶共振条件:$ {K_1} = \sqrt { {g}/({{4 T}})} , $ 和(46)式随时间线性增长的共振特解:
$ {B_{4 - 1 - 1 - 2}} = \pm {\text{i}}t{C_{4 - 1 - 1 - 2}}{{\text{e}}^{ \mp 4{\text{i}}{n_1}t}}. $ 其中,
$ {C_{4 - 1 - 1 - 2}} = {{{C_{4 - 1 - 1 - 1,2}}}}/({{8{n_1}}}). $ 同样, 可求出第四阶在另一对正负波数矢量点
$\left( { \pm 2{{\boldsymbol{K}}_1}} \right)\left( {3, 4} \right)$ 上的非共振和共振的自由表面位移的Fourier系数. 于是, 由上述第二、三、四阶自由表面位移的Fourier系数可求出各阶非共振和共振的自由表面位移，进而求出第三、四阶在各自不同对正负波数矢量点处的若干典型组合位移，最终得出单波列从第一阶到第二阶直至第四阶的非共振和共振的若干自由表面总位移. -
单波列的第二、第三、第四阶自共振表达式已依次推出, 它们均表现出鲜明的一致特征. 据此, 可顺势推断出单波列的第
$n$ 阶自共振定律:若海洋深水表面张力波-重力波的单波列自第二阶起始的第
$\left( {n - 1} \right)$ 阶波均不发生自共振, 则第$n$ 阶波发生自共振的条件为$ {K_1} = \sqrt { {g}/({{nT}})} \text{, }~~~~ n \geqslant 2. $ 令
$ \kappa = {1}/{n} = {{T{{\left( {{K_1}} \right)}^2}}}/{g}. $ 其中,
$\kappa $ 表示表面张力与重力的比值, 即著名的Bond数的倒数.依据深水海洋表面波的下列划分范围[27]: 表面张力波,
$ n < 0.5 $ ; 表面张力波-重力波,$0.5 < n < 6.45$ ; 表面重力波,$n > 6.45$ , 可给出单波列第$n$ 阶自共振定律(53)式的演变特征图1. 从中可知, 共振波数从“表面张力波”区域急剧地下滑到“表面张力波-重力波”过渡区域, 然后平缓地进入“表面重力波”区域. 在“表面张力波-重力波”区域内可发生5种正整数阶共振:$ n = 2, 3, 4, 5, 6 $ ; 在“表面重力波”区域内可发生无穷多正整数阶共振:$ n = 7, 8, \cdot \cdot \cdot $ . 在“表面张力波”区域内则无共振, 故以虚线表之. 但是, 纯表面张力波当可发生最低阶之第二阶的3-波共振[33]. 显见, 上述划分的“表面张力波和表面重力波”并非纯粹, 只是各以其主打“表面张力波-重力波”中的某种波. 若从Navier-Stokes方程出发, 则可发现存在一个无穷可数“波象(wave-like)”共振集合, 可同胚于Cantor有理数集合[39]. (53)式所包容的大于1的无穷多正整数阶共振集合便为其一个特定、典型的子集合, 也可更为深刻、通透地以“波粒二象性(wave-particle duality)”加以刻画[40].
图 1 深水海洋表面张力波-重力波的单波自共振定律, ρ = 1000 kg/m3,
$g = 9.81{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{m}}/{{\text{s}}^2}$ ,$T' = 0.074{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {{\text{N}}} /{{\text{m}}} $ ,$T = 7.4 \times $ $ {10^{ - 5}}~{{{\text{m}}} ^3}/{{{\text{s}}} ^2}$ Figure 1. Self-resonance law of one wave for ocean surface waves in deep water:
$\rho = 1000\;{{\text{kg}}}/{{{\text{m}}} ^{3}}$ ,$g = 9.81{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{m}}/{{\text{s}}^2}$ ,$T' = $ $ 0.074~{{\text{N}}} /{{\text{m}}}$ ,$T = 7.4 \times {10^{ - 5}}~{{{\text{m}}} ^3}/{{{\text{s}}} ^2}$ .将自由表面位移的动力学边界条件方程作某种典型级数展开, 将会发现在其各项的分母中出现幂乘积因子
$\left( {\kappa n - 1} \right)$ [41]. 这就意味着存在一种可数无穷的内在共振族[27], 也就给予共振定律(53)式一个充分证实. 从(53)式中可以看出: 当$n \to \infty $ 时,${K_1} \to 0$ . 这就预示着“物极必反”: 超大波长的波平浪静, 或海流. 所谓湖泊, 所谓海洋, 其动力学乃至于动理学的极致表现, 大致在此.在理论上, 可继续对(53)式的第五、第六阶直至第
$n$ 阶单波列持续进行一番循序渐进、愈来愈繁重的运算、求解、论证. 从中, 自可或可显出自共振定律(53)始终如一的特性. 在实验上, 可以精巧的构想、布局而证实(53)式. 例如, 可通过对重力环境的改变以达到表面张力和重力区域之间的理论变迁频率[22]. 的确, 因推演、推断或猜想而一举成功的著名物理学定律、方程, 总是数学物理或理论物理或应用数学向前推进的一座座不可或缺的里程碑. 例如, 石破天惊的牛顿运动定律、万有引力定律[42]和杨-米尔斯方程[43]. -
水波, 体现在色散性, 重在与其他波动共存的共振性. 以此, 可将能量从低阶向高阶显著地传播、扩散下去, 则最基本的单波列自共振一般特征不可不洞察. 本文按照经典的Phillips共振理论推导出单波列第二、三、四阶共振关系而从中推断出的海洋深水表面张力波-重力波的单波列自共振定律(53)式, 即是一大典型实例. 这就为深水海洋表面波的单重、多重[27]的双波列[25]、多波列[26]共振条件定律的提出奠定了基石, 而基本的对称性[44,45]必蕴含在这些定律的构造中. 从更为实际的波浪环境条件来看, 共振定律(53)式受制于理想的深水波. 若是较为实际的有限常水深波[46,47]呢? 更进一步的缓变[48]或陡变[49]的海底波呢? 再将环境流的作用纳入其中而包含基本、普遍的波-流-海底共振相互作用机制[50]呢? 无疑, (53)式可以充当这一切扩展的出发点、向导, 进而积极推演, 大胆预测, 最终为与波-流-海底共振密切关联的能量海浪谱和具有极大普适性的波湍流之Kolmogorov-Zakharov-型谱[3,14-16,19-22]的深入、完整研制提供一个可靠的理论基础平台.
-
波-波共振机制, 无论在微观物质还是在宏观物质的能量传播、分布进程中都起着一种根本、突显的作用. 对于地球上最为广阔、直观的海洋表面波运动, 势必更加如此而可观可知. 那么, 可否从中提炼出一般的波-波共振规律? 尤其是最为特殊、简要的单波列共振法则. 为此, 依据Phillips开创现代水波动力学而提出特定4-波共振条件的经典一整套方式方法, 从基本的海洋深水表面张力波-重力波控制方程组出发, 运用Fourier-Stieltjes变换和摄动方法依次给出自由表面位移的Fourier分量的第一阶微分方程和愈来愈复杂却趋于完整的第二、三、四阶积分微分方程, 在一套自创、自明而又简洁的符号体系下依次求解这些方程而求得其单波列第一阶自由表面位移和第二、三、四阶非共振与共振自由表面位移的Fourier系数以及第二、三、四阶共振条件, 从而顺势推断出一般的单波列第
$n$ 阶自共振定律. 这就完整揭示了海洋表面张力波-重力波之单波列共振动力学的丰富内涵, 有效扩展了海洋表面重力波之经典Phillips共振单波列解的适用范围, 为刻画海洋表面波之双波列、更多波列的单重、多重共振相互作用机制奠定了基石, 因而在全部波动领域的单波列共振规律的探寻上提供了一种典型范例.-
关键词:
- 自共振定律 /
- 单波列 /
- 深水表面张力波-重力波 /
- Fourier-Stieltjes变换
Wave-wave resonance mechanism plays a fundamental and prominent role in the process of energy transfer and distribution, whether it is in microscopic or macroscopic matter. For the most extensive and intuitive ocean surface wave motion on earth, it is bound to be even more so. Can we extract the general wave-wave resonance law from it, especially the most special and brief resonance law for single wave train? To this end, according to a set of classical methods proposed by Phillips for initiating modern water wave dynamics with the specific 4-wave resonance conditions, and starting from the basic governing equations of ocean deep-water surface capillary-gravity waves, the first-order differential equation, and the second-, third- and fourth-order integral differential ones, which are becoming more and more complex but tend to be complete, of the Fourier components of free surface displacement are respectively given by the Fourier-Stieltjes transformation and perturbation method. Under a set of symbol system, which is self-created, self-evident and concise, these equations are solved in turn to obtain the first-order free surface displacement of single wave train, the Fourier coefficients of the second-, third- and fourth-order non-resonant and resonant free surface displacements, and the second-, third- and fourth-order resonant conditions, thus leading to the general nth-order self-resonance law of single wave train. This completely reveals the rich connotation of single wave resonance dynamics of ocean surface capillary-gravity waves, effectively expands the application range of the classical single wave resonance solutions given by Phillips for ocean surface gravity waves, lays the foundation for depicting single and multiple resonance interaction mechanisms of double and multi-wave trains of ocean surface waves, and so provides a typical example for the exploration of single-wave resonance law in all wave fields.-
Keywords:
- self-resonance law /
- single wave train /
- deep-water capillary-gravity waves /
- Fourier-Stieltjes transform
[1] Whitham G B 1974 Linear and Nonlinear Waves (New York: Wiley) pp2–4
[2] Phillips O M 1960 J. Fluid Mech. 9 193
Google Scholar
[3] Hasselmann K 1962 J. Fluid Mech. 12 481
Google Scholar
[4] Longuet-Higgins M S 1962 J. Fluid Mech. 12 321
Google Scholar
[5] Benney D J 1962 J. Fluid Mech. 14 577
Google Scholar
[6] Bretherton F B 1964 J. Fluid Mech. 20 457
Google Scholar
[7] Longuet-Higgins M S, Smith N D 1966 J. Fluid Mech. 25 417
Google Scholar
[8] McGoldrick L F, Phillips O M, Huang N E, Hodgson T H 1966 J. Fluid Mech. 25 437
Google Scholar
[9] Sun C, Jia S, Barsi C, Rica S, Picozzi A, Fleischer J W 2012 Nat. Phys. 8 470
Google Scholar
[10] Dyachenko S, Newell A C, Pushkarev A, Zakharov V E 1992 Phys. D 57 96
[11] Nazarenko S, Lukaschuk S 2016 Annu. Rev. Condens. Matter. 7 61
Google Scholar
[12] Davis G, Jamin T, Deleuze J, Joubaud S, Dauxois T 2020 Phys. Rev. Lett. 124 204502
Google Scholar
[13] Galtier S, Nazarenko S V 2017 Phys. Rev. Lett. 119 221101
Google Scholar
[14] Zakharov V E, L’vov V S, Falkovich G 1992 Kolmogorov Spectra of Turbulence I: Wave Turbulence (Berlin: Springer-Verlag )
[15] Nazarenko S 2011 Wave Turbulence (Berlin: Springer)
[16] Newell A C, Rumpf B 2011 Annu. Rev. Fluid Mech. 43 59
Google Scholar
[17] 黄虎 2013 物理学报 62 139201
Google Scholar
Huang H 2013 Acta Phys. Sin. 62 139201
Google Scholar
[18] Krasitskii V P 1994 J. Fluid Mech. 272 1
Google Scholar
[19] Dyachenko A I, Korotkevich A O, Zakharov V E 2004 Phys. Rev. Lett. 92 134501
Google Scholar
[20] Griffin A, Krstulovic G, L’vov V S, Nazarenko S 2022 Phys. Rev. Lett. 128 224501
Google Scholar
[21] Dias F, Kharif C 1999 Annu. Rev. Fluid Mech. 31 301
Google Scholar
[22] Cazaubiel A, Mawet S, Darras A, Grosjean G, van Loon J J W A, Dorbolo S, Falcon E 2019 Phys. Rev. Lett. 123 244501
Google Scholar
[23] Aubourg Q, Mordant N 2015 Phys. Rev. Lett. 114 144501
Google Scholar
[24] Aubourg Q, Mordant N 2016 Phys. Rev. Fluids 1 023701
Google Scholar
[25] Madsen P A, Fuhrman D R 2006 J. Fluid Mech. 557 369
Google Scholar
[26] Madsen P A, Fuhrman D R 2012 J. Fluid Mech. 698 304
Google Scholar
[27] Hammack J L, Henderson D M 1993 Annu. Rev. Fluid Mech. 25 55
Google Scholar
[28] Stokes G G 1847 Trans. Camb. Phil. Soc. 8 441
[29] 崔巍, 闫在在, 木仁 2014 物理学报 63 140301
Google Scholar
Cui W, Yan Z Z, Mu R 2014 Acta Phys. Sin. 63 140301
Google Scholar
[30] Gowers T 主编 (齐民友 译)2014 普林斯顿数学指南 (北京: 科学出版社) 第333—334页
Gowers T (translated by Qi M Y) 2014 The Princeton Companion to Mathematics (Beijing: Science Press) pp333–334 (in Chinese)
[31] 梅凤翔 2003 物理学报 52 1048
Google Scholar
Mei F X 2003 Acta Phys. Sin. 52 1048
Google Scholar
[32] Zakharov V E 1968 J. Appl. Mech. Tech. Phys. 9 86
[33] McGoldrick L F 1965 J. Fluid Mech. 21 305
Google Scholar
[34] Krasitskii V P, Kozhelupova N G 1995 Oceanology 34 435
[35] Lin G B, Huang H 2019 China Ocean Eng. 33 734
Google Scholar
[36] 老子 2014 老子 (北京: 中华书局) 第165页
Lao Z 2014 Lao Zi (Beijing: Zhonghua Book Company) p165 (in Chinese)
[37] Bender C M, Orszag S A 1978 Advanced Mathematical Methods for Scientists and Engineers (Berlin: Springer)
[38] 马召召, 杨庆超, 周瑞平 2021 物理学报 70 240501
Google Scholar
Ma Z Z, Yang Q C, Zhou R P 2021 Acta Phys. Sin. 70 240501
Google Scholar
[39] Yao L S 1999 J. Fluid Mech. 395 237
Google Scholar
[40] Hasselmann K 1963 J. Fluid Mech. 15 273
Google Scholar
[41] Wilton J R 1915 Phil. Mag. 29 688
Google Scholar
[42] 牛顿 (王可迪 译) 2006 自然哲学之数学原理 (北京: 北京大学出版社)
Newton I (translated by Wang K D) 2006 Mathematical Principles of Natural Philosophy (Beijing: Peking University Press) (in Chinese)
[43] Yang C N, Mills R L 1954 The Phys. Rev. 96 191
Google Scholar
[44] Marsden J E, Ratiu T S 1999 Introduction to Mechanics and Symmetry (Berlin: Springer )
[45] 叶鹏 2020 物理学报 69 077102
Google Scholar
Ye P 2020 Acta Phys. Sin. 69 077102
Google Scholar
[46] Matsuno Y 1992 Phys. Rev. Lett. 69 609
Google Scholar
[47] 黄虎, 夏应波 2011 物理学报 60 044702
Google Scholar
Huang H, Xia Y B 2011 Acta Phys. Sin. 60 044702
Google Scholar
[48] 黄虎 2010 物理学报 59 740
Google Scholar
Huang H 2010 Acta Phys. Sin. 59 740
Google Scholar
[49] Artiles W, Nachbin A 2004 Phys. Rev. Lett. 93 234501
Google Scholar
[50] Huang H 2009 Dynamics of Surface Waves in Coastal Waters: Wave-Current-Bottom Interactions (Beijing, Berlin: Higher Education Press, Springer)
-
图 1 深水海洋表面张力波-重力波的单波自共振定律, ρ = 1000 kg/m3,
$g = 9.81{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{m}}/{{\text{s}}^2}$ ,$T' = 0.074{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {{\text{N}}} /{{\text{m}}} $ ,$T = 7.4 \times $ $ {10^{ - 5}}~{{{\text{m}}} ^3}/{{{\text{s}}} ^2}$ Fig. 1. Self-resonance law of one wave for ocean surface waves in deep water:
$\rho = 1000\;{{\text{kg}}}/{{{\text{m}}} ^{3}}$ ,$g = 9.81{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{m}}/{{\text{s}}^2}$ ,$T' = $ $ 0.074~{{\text{N}}} /{{\text{m}}}$ ,$T = 7.4 \times {10^{ - 5}}~{{{\text{m}}} ^3}/{{{\text{s}}} ^2}$ . -
[1] Whitham G B 1974 Linear and Nonlinear Waves (New York: Wiley) pp2–4
[2] Phillips O M 1960 J. Fluid Mech. 9 193
Google Scholar
[3] Hasselmann K 1962 J. Fluid Mech. 12 481
Google Scholar
[4] Longuet-Higgins M S 1962 J. Fluid Mech. 12 321
Google Scholar
[5] Benney D J 1962 J. Fluid Mech. 14 577
Google Scholar
[6] Bretherton F B 1964 J. Fluid Mech. 20 457
Google Scholar
[7] Longuet-Higgins M S, Smith N D 1966 J. Fluid Mech. 25 417
Google Scholar
[8] McGoldrick L F, Phillips O M, Huang N E, Hodgson T H 1966 J. Fluid Mech. 25 437
Google Scholar
[9] Sun C, Jia S, Barsi C, Rica S, Picozzi A, Fleischer J W 2012 Nat. Phys. 8 470
Google Scholar
[10] Dyachenko S, Newell A C, Pushkarev A, Zakharov V E 1992 Phys. D 57 96
[11] Nazarenko S, Lukaschuk S 2016 Annu. Rev. Condens. Matter. 7 61
Google Scholar
[12] Davis G, Jamin T, Deleuze J, Joubaud S, Dauxois T 2020 Phys. Rev. Lett. 124 204502
Google Scholar
[13] Galtier S, Nazarenko S V 2017 Phys. Rev. Lett. 119 221101
Google Scholar
[14] Zakharov V E, L’vov V S, Falkovich G 1992 Kolmogorov Spectra of Turbulence I: Wave Turbulence (Berlin: Springer-Verlag )
[15] Nazarenko S 2011 Wave Turbulence (Berlin: Springer)
[16] Newell A C, Rumpf B 2011 Annu. Rev. Fluid Mech. 43 59
Google Scholar
[17] 黄虎 2013 物理学报 62 139201
Google Scholar
Huang H 2013 Acta Phys. Sin. 62 139201
Google Scholar
[18] Krasitskii V P 1994 J. Fluid Mech. 272 1
Google Scholar
[19] Dyachenko A I, Korotkevich A O, Zakharov V E 2004 Phys. Rev. Lett. 92 134501
Google Scholar
[20] Griffin A, Krstulovic G, L’vov V S, Nazarenko S 2022 Phys. Rev. Lett. 128 224501
Google Scholar
[21] Dias F, Kharif C 1999 Annu. Rev. Fluid Mech. 31 301
Google Scholar
[22] Cazaubiel A, Mawet S, Darras A, Grosjean G, van Loon J J W A, Dorbolo S, Falcon E 2019 Phys. Rev. Lett. 123 244501
Google Scholar
[23] Aubourg Q, Mordant N 2015 Phys. Rev. Lett. 114 144501
Google Scholar
[24] Aubourg Q, Mordant N 2016 Phys. Rev. Fluids 1 023701
Google Scholar
[25] Madsen P A, Fuhrman D R 2006 J. Fluid Mech. 557 369
Google Scholar
[26] Madsen P A, Fuhrman D R 2012 J. Fluid Mech. 698 304
Google Scholar
[27] Hammack J L, Henderson D M 1993 Annu. Rev. Fluid Mech. 25 55
Google Scholar
[28] Stokes G G 1847 Trans. Camb. Phil. Soc. 8 441
[29] 崔巍, 闫在在, 木仁 2014 物理学报 63 140301
Google Scholar
Cui W, Yan Z Z, Mu R 2014 Acta Phys. Sin. 63 140301
Google Scholar
[30] Gowers T 主编 (齐民友 译)2014 普林斯顿数学指南 (北京: 科学出版社) 第333—334页
Gowers T (translated by Qi M Y) 2014 The Princeton Companion to Mathematics (Beijing: Science Press) pp333–334 (in Chinese)
[31] 梅凤翔 2003 物理学报 52 1048
Google Scholar
Mei F X 2003 Acta Phys. Sin. 52 1048
Google Scholar
[32] Zakharov V E 1968 J. Appl. Mech. Tech. Phys. 9 86
[33] McGoldrick L F 1965 J. Fluid Mech. 21 305
Google Scholar
[34] Krasitskii V P, Kozhelupova N G 1995 Oceanology 34 435
[35] Lin G B, Huang H 2019 China Ocean Eng. 33 734
Google Scholar
[36] 老子 2014 老子 (北京: 中华书局) 第165页
Lao Z 2014 Lao Zi (Beijing: Zhonghua Book Company) p165 (in Chinese)
[37] Bender C M, Orszag S A 1978 Advanced Mathematical Methods for Scientists and Engineers (Berlin: Springer)
[38] 马召召, 杨庆超, 周瑞平 2021 物理学报 70 240501
Google Scholar
Ma Z Z, Yang Q C, Zhou R P 2021 Acta Phys. Sin. 70 240501
Google Scholar
[39] Yao L S 1999 J. Fluid Mech. 395 237
Google Scholar
[40] Hasselmann K 1963 J. Fluid Mech. 15 273
Google Scholar
[41] Wilton J R 1915 Phil. Mag. 29 688
Google Scholar
[42] 牛顿 (王可迪 译) 2006 自然哲学之数学原理 (北京: 北京大学出版社)
Newton I (translated by Wang K D) 2006 Mathematical Principles of Natural Philosophy (Beijing: Peking University Press) (in Chinese)
[43] Yang C N, Mills R L 1954 The Phys. Rev. 96 191
Google Scholar
[44] Marsden J E, Ratiu T S 1999 Introduction to Mechanics and Symmetry (Berlin: Springer )
[45] 叶鹏 2020 物理学报 69 077102
Google Scholar
Ye P 2020 Acta Phys. Sin. 69 077102
Google Scholar
[46] Matsuno Y 1992 Phys. Rev. Lett. 69 609
Google Scholar
[47] 黄虎, 夏应波 2011 物理学报 60 044702
Google Scholar
Huang H, Xia Y B 2011 Acta Phys. Sin. 60 044702
Google Scholar
[48] 黄虎 2010 物理学报 59 740
Google Scholar
Huang H 2010 Acta Phys. Sin. 59 740
Google Scholar
[49] Artiles W, Nachbin A 2004 Phys. Rev. Lett. 93 234501
Google Scholar
[50] Huang H 2009 Dynamics of Surface Waves in Coastal Waters: Wave-Current-Bottom Interactions (Beijing, Berlin: Higher Education Press, Springer)
计量
- 文章访问数: 367
- PDF下载量: 17
- 被引次数: 0
