竞争线性和非线性衍射下的空间孤子
Spatial solitons under competing linear and nonlinear diffractions
摘要：
我们引入了一个一般模型，该模型通过与线性衍射竞争的非线性衍射项来扩充一维非线性薛定谔（NLS）方程。新的条款包含两个不可约的参数，并承认一个自然的光学介质形式的哈密顿表示。该方程可作为非线性光子晶体中超准直点附近空间孤子的模型。在此模型的框架下，详细分析了基本孤立波，包括变分近似（VA），精确的解析结果，和系统的数值计算。Vakhitov-Kolokolov（VK）准则被用来精确地预测孤子的稳定边界，这是在一个精确的分析形式，以及波可能拥有的最大总功率（范数）。超过临界点，观察到崩溃效应，由适当的扰动引起。利用直接数值模拟的方法研究了两束完全相同的平行孤立光束之间的相互作用。结果表明，随着非线性衍射强度的增加，同相孤子会合并成强脉冲或坍缩脉冲。
英文摘要：
We introduce a general model which augments the one-dimensional nonlinear Schrödinger (NLS) equation by nonlinear-diffraction terms competing with the linear diffraction. The new terms contain two irreducible parameters and admit a Hamiltonian representation in a form natural for optical media. The equation serves as a model for spatial solitons near the supercollimation point in nonlinear photonic crystals. In the framework of this model, a detailed analysis of the fundamental solitary waves is reported, including the variational approximation (VA), exact analytical results, and systematic numerical computations. The Vakhitov-Kolokolov (VK) criterion is used to precisely predict the stability border for the solitons, which is found in an exact analytical form, along with the largest total power (norm) that the waves may possess. Past a critical point, collapse effects are observed, caused by suitable perturbations. Interactions between two identical parallel solitary beams are explored by dint of direct numerical simulations. It is found that in-phase solitons merge into robust or collapsing pulsons, depending on the strength of the nonlinear diffraction.
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