发表机构
Bar-Ilan University（巴伊兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究有损非线性传输线的行波前沿，通过指定双曲正切剖面求解电压-电荷关系，构造近似解并分析原偏微分方程以区分压缩激波与非压缩扭结。
AI 中文摘要
我们研究有损非线性传输线中的行波前沿，通过指定无量纲宽度为$m$的双曲正切剖面，求解使该剖面成为精确解的电压-电荷关系这一反问题，该精确关系用高斯超几何函数表示。我们还通过假设立方电压-电荷关系，并将得到的二次阻尼函数替换为其最优线性最小二乘近似，构造近似解，该构造确定了立方系数并得到修正的控制方程，该方程允许指定剖面作为精确解。在两种构造中，物理可容许性选择激波前沿；在宽前沿极限下，近似归一化力在$m^{-2}$阶与精确结果一致，数值比较表明，即使在大$m$的渐近区域外，一致性仍良好。最后，对原偏微分方程的特征分析无需借助力学类比即可区分压缩激波与非压缩扭结。
英文摘要
We consider a lossy transmission line with a nonlinear voltage--charge relation. We derive an equation for a traveling front with the charge approaching constant asymptotic values on both sides of the front and solve the inverse problem for this equation exactly. Starting from a prescribed monotonic front profile and a prescribed front speed, we determine the local sound speed within the front. This quantity is the central object of our analysis and allows us to determine the voltage--charge relation of the transmission line in which the front propagates. The squared sound speed, averaged uniformly over the charge interval spanned by the front, is equal to the squared front speed. We specifically consider fronts with a power-logistic profile and its special case, the hyperbolic-tangent profile. All physically admissible fronts of this form are shocks rather than kinks. The voltage--charge relation of the transmission line in which the power-logistic shocks propagate is expressed in terms of the lower incomplete beta function. We propose an approximate method of solving the traveling front equation in a transmission line with the prescribed voltage--charge relation by the inverse-design approach. We examine the local spectral properties of the front in terms of the original partial differential equations and show that the short-wavelength perturbation modes are damped. The kink--shock distinction is considered in the framework of the long-wavelength characteristic analysis.
CommentspdfLaTeX, 10 pages, 3 Figures. The manuscript has undergone major revision. Accepted for publication in Mathematics
