Nikitenkova S, Singh N, Stepanyants Y
Lobachevsky State University of Nizhny Novgorod, Russia.
University of Southern Queensland, West St., Toowoomba, Queensland 4350, Australia.
Chaos. 2015 Dec;25(12):123113. doi: 10.1063/1.4937362.
In this paper, we revisit the problem of modulation stability of quasi-monochromatic wave-trains propagating in a media with the double dispersion occurring both at small and large wavenumbers. We start with the shallow-water equations derived by Shrira [Izv., Acad. Sci., USSR, Atmos. Ocean. Phys. (Engl. Transl.) 17, 55-59 (1981)] which describes both surface and internal long waves in a rotating fluid. The small-scale (Boussinesq-type) dispersion is assumed to be weak, whereas the large-scale (Coriolis-type) dispersion is considered as without any restriction. For unidirectional waves propagating in one direction, only the considered set of equations reduces to the Gardner-Ostrovsky equation which is applicable only within a finite range of wavenumbers. We derive the nonlinear Schrödinger equation (NLSE) which describes the evolution of narrow-band wave-trains and show that within a more general bi-directional equation the wave-trains, similar to that derived from the Ostrovsky equation, are also modulationally stable at relatively small wavenumbers k < kc and unstable at k > kc, where kc is some critical wavenumber. The NLSE derived here has a wider range of applicability: it is valid for arbitrarily small wavenumbers. We present the analysis of coefficients of the NLSE for different signs of coefficients of the governing equation and compare them with those derived from the Ostrovsky equation. The analysis shows that for weakly dispersive waves in the range of parameters where the Gardner-Ostrovsky equation is valid, the cubic nonlinearity does not contribute to the nonlinear coefficient of NLSE; therefore, the NLSE can be correctly derived from the Ostrovsky equation.
在本文中，我们重新审视了在小和大的波数下均存在双色散的介质中传播的准单色波列的调制稳定性问题。我们从什里拉[《苏联科学院通报，大气与海洋物理学报》（英文版）17, 55 - 59 (1981)]推导的浅水方程出发，该方程描述了旋转流体中的表面长波和内长波。假设小尺度（布辛涅斯克型）色散较弱，而大尺度（科里奥利型）色散不受任何限制。对于沿一个方向传播的单向波，只有所考虑的方程组会简化为仅适用于有限波数范围的加德纳 - 奥斯特罗夫斯基方程。我们推导了描述窄带波列演化的非线性薛定谔方程（NLSE），并表明在更一般的双向方程中，类似于从奥斯特罗夫斯基方程导出的波列，在相对较小的波数(k < k_c)时也是调制稳定的，而在(k > k_c)时是不稳定的，其中(k_c)是某个临界波数。这里推导的NLSE具有更广泛的适用范围：它对任意小的波数都是有效的。我们针对控制方程系数的不同符号对NLSE的系数进行了分析，并将它们与从奥斯特罗夫斯基方程导出的系数进行了比较。分析表明，在加德纳 - 奥斯特罗夫斯基方程有效的参数范围内，对于弱色散波，三次非线性对NLSE的非线性系数没有贡献；因此，NLSE可以从奥斯特罗夫斯基方程正确推导出来。
