本文通过把十二个Jacobi椭圆函数分类成四组,从而提出一个新的广义Jacobi椭圆函数展开法来构造非线性演化方程的精确双周期解。在数学软件Maple的帮助下,应用这个非常有效的方法求出了非线性演化方程的许多解,当模数m→0或1时,这些解退化为相应的孤立波解或三角函数解。In this letter, twelve Jacobi elliptic functions are divided into four groups, and a new general Jacobi elliptic function expansion method is proposed to construct abundant exact doubly periodic solutions of nonlinear evolution equations. As a result, with the aid of computer symbolic computation software (for example, Maple), many exact doubly periodic solutions are obtained which shows that this method is very powerful. When the modulus m→0 or 1, these solutions degenerate to the corresponding solitary wave solutions and trigonometric function (singly periodic) solutions.
一个新的广义的Jacobi椭圆函数有理展开法被提出来构造非线性波动方程的有理解。利用这个直接有效的方法,获得了许多关于Jacobi椭圆函数的有理解。当模数m→0或1时,这些解退化为相应的关于孤立波或三角函数的有理解。A new general Jacobi elliptic function rational expansion procedure is presented for constructing rational solutions of nonlinear wave equations in terms of the Jacobi elliptic function. As a consequence, many new rational form Jacobi elliptic function solutions are obtained by this powerful and direct method. Moreover, the corresponding rational form solitary wave solutions and rational form trigonometric function solutions are also obtained when the modulus m→0 or 1.
建立了改进的基于Jacobi椭圆函数的随机平均法,用于预测有界噪声激励作用下硬弹簧和软弹簧系统的随机响应.通过引入基于Jacobi椭圆函数的变换,导出关于响应幅值和激励与响应之间相位差的随机微分方程,应用随机平均原理,将响应幅值近似为一个Markov扩散过程,建立其平均的Ito随机微分方程.响应幅值的稳态概率密度由相应的简化Fokker-Planck-Kolmogorov方程解出;进而得到系统位移和速度的稳态概率密度.以Duffing-Van der Pol振子为例,研究了硬刚度及软刚度情形下的随机响应,通过与Monte Carlo数值模拟结果比较证实了此方法的可行性及精度.由于广义调和函数是基于线性系统的精确解,Jacobi椭圆函数是基于非线性系统的精确解,研究结果表明基于Jacobi椭圆函数的随机平均法得到的结果与Monte Carlo模拟方法更接近.因此与基于广义调和函数的随机平均相比,基于Jacobi椭圆函数更加精确,因为它是基于保守的非线性系统.
