$(2+1)-$维中的超短脉冲和短脉冲方程
Ultrashort pulses and short-pulse equations in $(2+1)-$dimensions
摘要：
本文导出并研究了$(2+1)-$维中的两种形式的短脉冲方程。使用麦克斯韦方程组作为出发点，以及介质的介电常数和磁导率的合适的Kramers-Kronig公式，这些公式是相关的，例如，对于左手异向介质和介质平板波导，我们采用多尺度技术得到了相应的模型。所产生的$(2+1)$维SPE的一般性质，包括基本的守恒定律，以及拉格朗日和哈密顿结构和数值模拟的一维和二维的初始数据，提出。超短1D呼吸似乎是相当强大的，而相当一般的二维本地化的初始条件被转换成准一维分散波形。
英文摘要：
In this paper, we derive and study two versions of the short pulse equation (SPE) in $(2+1)-$dimensions. Using Maxwell's equations as a starting point, and suitable Kramers-Kronig formulas for the permittivity and permeability of the medium, which are relevant, e.g., to left-handed metamaterials and dielectric slab waveguides, we employ a multiple scales technique to obtain the relevant models. General properties of the resulting $(2+1)$-dimensional SPEs, including fundamental conservation laws, as well as the Lagrangian and Hamiltonian structure and numerical simulations for one- and two-dimensional initial data, are presented. Ultrashort 1D breathers appear to be fairly robust, while rather general two-dimensional localized initial conditions are transformed into quasi-one-dimensional dispersing waveforms.
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