Chow K W, Merhasin Ilya M, Malomed Boris A, Nakkeeran K, Senthilnathan K, Wai P K A
Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Feb;77(2 Pt 2):026602. doi: 10.1103/PhysRevE.77.026602. Epub 2008 Feb 7.
We construct two families of exact periodic solutions to the standard model of fiber Bragg grating (FBG) with Kerr nonlinearity. The solutions are named "sn" and "cn" waves, according to the elliptic functions used in their analytical representation. The sn wave exists only inside the FBG's spectral bandgap, while waves of the cn type may only exist at negative frequencies (omega<0), both inside and outside the bandgap. In the long-wave limit, the sn and cn families recover, respectively, the ordinary gap solitons, and (unstable) antidark and dark solitons. Stability of the periodic solutions is checked by direct numerical simulations and, in the case of the sn family, also through the calculation of instability growth rates for small perturbations. Although, rigorously speaking, all periodic solutions are unstable, a subfamily of practically stable sn waves, with a sufficiently large spatial period and omega>0, is identified. However, the sn waves with omega<0, as well as all cn solutions, are strongly unstable.
我们构造了具有克尔非线性的光纤布拉格光栅（FBG）标准模型的两个精确周期解族。根据其解析表示中使用的椭圆函数，这些解被命名为“sn”波和“cn”波。sn波仅存在于FBG的光谱带隙内，而cn型波可能仅存在于带隙内外的负频率（ω<0）处。在长波极限下，sn族和cn族分别恢复为普通带隙孤子以及（不稳定的）反暗孤子和暗孤子。通过直接数值模拟检查周期解的稳定性，对于sn族，还通过计算小扰动的不稳定性增长率来检查。尽管严格来说，所有周期解都是不稳定的，但我们确定了一个具有足够大空间周期且ω>0的实际稳定sn波子族。然而，ω<0的sn波以及所有cn解都是强不稳定的。
