论文标题
分散介质中高频传播的近似
Approximation of high-frequency wave propagation in dispersive media
论文作者
论文摘要
我们考虑具有三线性非线性的半线性双曲线系统。微分方程和初始数据都包含一个小参数$ \ varepsilon $的倒数，并且典型的解决方案在时间和空间中以频率成比例为$ 1/\ varepsilon $振荡。此外，必须按时间间隔计算$ 1/\ varepsilon $的时间间隔，以研究非线性和衍射效应。结果，直接的数值模拟非常昂贵甚至是不可能的。我们提出了一个分析近似值，并证明它将精确的解决方案近似于$ \ Mathcal {o}的错误（\ Varepsilon^2）$ on Length的时间间隔$ 1/\ varepsilon $。这是对经典非线性Schrödinger近似的重大改进，该近似仅产生$ \ Mathcal {O}（\ Varepsilon）$的精度。
We consider semilinear hyperbolic systems with a trilinear nonlinearity. Both the differential equation and the initial data contain the inverse of a small parameter $\varepsilon$, and typical solutions oscillate with frequency proportional to $1/\varepsilon$ in time and space. Moreover, solutions have to be computed on time intervals of length $1/\varepsilon$ in order to study nonlinear and diffractive effects. As a consequence, direct numerical simulations are extremely costly or even impossible. We propose an analytical approximation and prove that it approximates the exact solution up to an error of $\mathcal{O}(\varepsilon^2)$ on time intervals of length $1/\varepsilon$. This is a significant improvement over the classical nonlinear Schrödinger approximation, which only yields an accuracy of $\mathcal{O}(\varepsilon)$.
