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专业: 凝聚态物性 II ：电子结构

[求助] 如题，电报方程引入非线性微扰后该如何解答，紧急求助

2. Consider a lossless ideal transmission line with inductance per unit length L, and
capacitance per unit length C. The current and voltage distribution is V(x,t) and
I(x,t) taken  together,  these  two  parameters  determine  the  complete  electromagnetic pattern in the system.  Energy propagates in the system due to the telegrapher’s equations:
∂V = −L ∂I
∂x       ∂t                                                                                        (2.1)

∂I = −C ∂V
∂x          ∂t                                                    (2.2)

At  x=-‐∞,  an  RF  source  produces  an  excitation  with  amplitude  1  and  angular frequency ω. We’ll consider this RF source to be an ideal, unidirectional source; that is, any incident radiation traveling in the negative x direction will not be reflected,but continue to ‐∞. Thus the system has the solution:
V(x,t)=cos(kx-wt)                                                             (2.3)
Here we have k =  (LC)^0.5  ω                                                                   (2.4)
Now consider the possibility that the capacitance on the region (0,L) becomes slightly nonlinear. That is to say, the relationship between the charge per unit length q, and the voltage V, becomes: q = CV − δV 2  (2.5)
Here δ is a small perturbation; we have δV< Note that if δ were 0, then the system would simply be that described by (2.1) and
(2.2) and (2.3) would be the solution.
Calculate the new solution for V as a function of x and t for the modified system to first order in δ.

我的想法是总有V的时间和空间二次偏微分方程，存在微扰后C'=(qV'-δV' 2)/V' =q- 2δV'，V’为单位长度两端电势差。取原电势为 v ，V ' =v+U ，U 作为微扰项远小于 v，有C'近似为q-2δv,对于v ，在固定单位区间上是含时的吧。。。。。不知道怎么处理。........还有我的理解有误吗

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