High-Speed Transmission in Long-Haul Electrical Systems

Abstract

We study the equations governing the high-speed transmission in long-haul electrical systems $i{\partial }_{t}u-\left(\mathrm{1}/\mathrm{3}\right){\left|{\partial }_x\right|}^{\mathrm{3}}u=i\lambda {\partial }_x\left({\left|u\right|}^{\mathrm{2}}u\right)$, $\left(t,x\right)\in {\mathbb{R}}^{+}\times \mathbb{R}$, $u\left(\mathrm{0},x\right)=u_{\mathrm{0}}\left(x\right)$, $x\in \mathbb{R},$ where $\lambda \in \mathbb{R},{\left|{\partial }_x\right|}^{\alpha }={\mathcal{F}}^{-\mathrm{1}}{\left|\xi \right|}^{\alpha }\mathcal{F}$, and $\mathcal{F}$ is the Fourier transformation. Our purpose in this paper is to obtain the large time asymptotics for the solutions under the nonzero mass condition $\int u_{\mathrm{0}}\left(x\right)dx\ne \mathrm{0}.$