AI 中文总结
本文用变分与混合变分近似研究空间工程CQ非线性波导中的孤子稳定性及碰撞动力学，揭示两类特征曲线和四种碰撞结果，以低计算成本有效描述复杂孤子相互作用。
AI 中文摘要
我们利用变分近似（VA）、混合变分近似（HVA）和直接数值模拟，研究了具有空间工程折射率和竞争性三次-五次（CQ）非线性分布的非线性光学介质中的空间孤子及其相互作用动力学。该模型在对称阶跃折射率平面介质波导中实现，其纤芯具有竞争性CQ非线性，而包层仅具有三次非线性响应。对于稳态，高斯VA预测了两种类型的$N(\mu)$特征曲线，其中$N$为孤子模数，$\mu$为传播常数，这两种曲线在参数空间中由一边界面分隔，数值计算揭示了相同的两种类型。在该边界面以下，VA与数值结果主要在低功率下吻合，而在高功率下，轮廓和稳定性出现显著偏差。在该边界面以上，变分曲线和数值曲线保持相同的定性形式，且超高斯拟设能准确描述高功率平顶孤子。孤子碰撞产生四种相互作用后状态：振荡态、分子态、分裂态和破裂态。HVA在宽功率范围内重现了前三种状态，包括涉及平顶孤子的碰撞。其主要局限性出现在破裂态，此时强辐射离开导波区域，无法由所采用的HVA拟设表示。然而，对于与第二类特征曲线相关的孤子，HVA仍能定性地捕捉破裂动力学。因此，与直接数值模拟相比，HVA能以显著更低的计算成本提供对复杂孤子相互作用的有效描述。
英文摘要
We study spatial solitons and their interaction dynamics in nonlinear optical media with spatially engineered refractive index and competing cubic-quintic (CQ) nonlinear profiles, using the variational approximation (VA), the hybrid variational approximation (HVA), and direct numerical simulations. The model was implemented in a symmetric step-index planar dielectric waveguide with a core exhibiting competing CQ nonlinearity and cladding layers possessing only a cubic nonlinear response. For stationary states, the Gaussian VA predicts two types of $N(μ)$ characteristic curves, where $N$ is soliton norm and $μ$ is propagation constant, separated by a boundary surface in parameter space, and numerical calculations reveal the same two types. Below this surface, the VA agrees with the numerical results mainly at low powers, while pronounced deviations in the profile and stability appear at high powers. Above the surface, the variational and numerical curves retain the same qualitative form, and a super-Gaussian ansatz accurately describes the high-power flat-top solitons. Soliton collisions produce four post-interaction regimes: Oscillation, Molecular, Splitting, and Breakup. The HVA reproduces the first three over a broad power range, including collisions involving flat-top solitons. Its main limitation arises in the Breakup regime, where strong radiation leaves the guiding region and cannot be represented by the adopted HVA ansatz. Nevertheless, for solitons associated with the second type of characteristic curves, the HVA still captures the breakup dynamics qualitatively. Thus, the HVA provides an efficient description of complex soliton interactions at a substantially lower computational cost than direct numerical simulations.
Comments12 pages, 11 figures, submitted for publication
