Discrete embedded solitons
Discrete embedded solitons
复制标题
离散嵌入孤子
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
B. Malomed
中科院分区：
文献类型：
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作者：
K. Yagasaki;A. Champneys;B. Malomed
研究了非线性动力学晶格模型中声子带内的离散嵌入孤子（ESs）的存在性及其性质。该模型描述了具有χ（2）（二次谐波产生）和χ（3）（克尔）非线性的一维光波导阵列，已知在连续极限中出现丰富的ES族。首先，考虑一个简单的激励问题，其中χ（3）非线性作用在单个波导中。在大波数极限下渐近地构造了一个显式解。一般的问题，然后被证明是等价的存在一个同宿轨道的四维可逆映射。从这些映射的性质，它表明（不同于普通的间隙孤子）离散ES具有相同的余维作为他们的连续对应。一个特定的数值方法来计算同宿解的地图，是对称的一个特定的反向变换。存在性，然后研究在全参数空间的问题。数值结果在适当的极限下与渐近结果一致，并表明离散ES可能与连续ES一样是半稳定的。
We address the existence and properties of discrete embedded solitons (ESs), that is, localized waves existing inside the phonon band in a nonlinear dynamical-lattice model. The model describes a one-dimensional array of optical waveguides with both χ(2) (second-harmonic generation) and χ(3) (Kerr) nonlinearities, for which a rich family of ESs are known to occur in the continuum limit. First, a simple motivating problem is considered, in which the χ(3) nonlinearity acts in a single waveguide. An explicit solution is constructed asymptotically in the large wavenumber limit. The general problem is then shown to be equivalent to the existence of a homoclinic orbit in a four-dimensional reversible map. From properties of such maps, it is shown that (unlike ordinary gap solitons) discrete ESs have the same codimension as their continuum counterparts. A specific numerical method is developed to compute homoclinic solutions of the map, that are symmetric under a specific reversing transformation. Existence is then studied in the full parameter space of the problem. Numerical results agree with the asymptotic results in the appropriate limit and suggest that the discrete ESs may be semi-stable as in the continuous case.