线性4阶泊松-费米方程的解

4阶泊松-费米方程为:

\begin{equation} \frac{d^2\phi}{dx^2}-\delta_c^2\frac{d^4\phi}{dx^4}=\frac{\sinh \phi}{1+2\gamma \sinh^2 \phi/2}=\rho(\phi) \label{Poisson-Fermi4} \end{equation}

低电势极限下,$\phi \ll 1$,方程\eqref{Poisson-Fermi4}右边为 $\phi$,方程为

\begin{equation} \delta_c^2\frac{d^4\phi}{dx^4}-\frac{d^2\phi}{dx^2}+\phi=0 \label{LPoisson-Fermi4} \end{equation}

这是一个高阶常系数线性常微分方程,下面给出解析解。

特征多项式为:

\begin{equation} \delta_c^2\lambda^4-\lambda^2+1=0 \label{charpolynomial} \end{equation}

解此方程得

\begin{equation} \lambda^2=z=\frac{1\pm\sqrt{1-4\delta_c^2}}{2\delta_c^2} \label{lambda2} \end{equation}

当$\delta_c\lt 1/2$时,

\begin{equation} \begin{split} \lambda_1=&\sqrt{\frac{1+\sqrt{1-4\delta_c^2}}{2\delta_c^2}}\\ \lambda_2=&-\sqrt{\frac{1+\sqrt{1-4\delta_c^2}}{2\delta_c^2}}\\ \lambda_3=&\sqrt{\frac{1-\sqrt{1-4\delta_c^2}}{2\delta_c^2}}\\ \lambda_4=&-\sqrt{\frac{1-\sqrt{1-4\delta_c^2}}{2\delta_c^2}} \end{split} \label{lambda1case} \end{equation}

此时,方程\eqref{LPoisson-Fermi4}的通解为

\begin{equation} \phi(x)=C_1e^{\lambda_1x}+C_2e^{\lambda_2x}+C_3e^{\lambda_3x}+C_4e^{\lambda_4x} \label{solution1case} \end{equation}

当$\delta_c = 1/2$时,

\begin{equation} \begin{split} \lambda_1=\lambda_2=&\sqrt{2}\\ \lambda_3=\lambda_4=&-\sqrt{2} \end{split} \label{lambda2case} \end{equation}

此时,方程\eqref{LPoisson-Fermi4}的通解为

\begin{equation} \phi(x)=C_1e^{\sqrt{2}x}+C_2xe^{\sqrt{2}x}+C_3e^{-\sqrt{2}x}+C_4xe^{-\sqrt{2}x} \label{solution2case} \end{equation}

当$\delta_c \gt 1/2$时,

\begin{equation} \lambda^2=z=\frac{1\pm i\sqrt{4\delta_c^2-1}}{2\delta_c^2}=\frac{1}{\delta_c}e^{\pm i\varphi} (\tan\varphi=\sqrt{4\delta_c^2-1}) \label{lambda2i} \end{equation}

特征根:

\begin{equation} \begin{split} \lambda_1=&\frac{1}{\sqrt{\delta_c}}e^{i\varphi/2}=\frac{\sqrt{2\delta_c+1}}{2\delta_c}+i\frac{\sqrt{2\delta_c-1}}{2\delta_c}\\ \lambda_2=&-\frac{1}{\sqrt{\delta_c}}e^{i\varphi/2}=-\frac{\sqrt{2\delta_c+1}}{2\delta_c}-i\frac{\sqrt{2\delta_c-1}}{2\delta_c}\\ \lambda_3=&\frac{1}{\sqrt{\delta_c}}e^{-i\varphi/2}=\frac{\sqrt{2\delta_c+1}}{2\delta_c}-i\frac{\sqrt{2\delta_c-1}}{2\delta_c}\\ \lambda_4=&-\frac{1}{\sqrt{\delta_c}}e^{-i\varphi/2}=-\frac{\sqrt{2\delta_c+1}}{2\delta_c}+i\frac{\sqrt{2\delta_c-1}}{2\delta_c} \end{split} \label{lambda3case} \end{equation}

此时,方程\eqref{LPoisson-Fermi4}的通解为

\begin{equation} \begin{split} \phi(x)=&C_1e^{\frac{\sqrt{2\delta_c+1}}{2\delta_c}x}\cos\left( \frac{\sqrt{2\delta_c-1}}{2\delta_c}x\right)+C_2e^{\frac{\sqrt{2\delta_c+1}}{2\delta_c}x}\sin\left( \frac{\sqrt{2\delta_c-1}}{2\delta_c}x\right)+\\ &C_3e^{-\frac{\sqrt{2\delta_c-1}}{2\delta_c}x}\cos\left( \frac{\sqrt{2\delta_c+1}}{2\delta_c}x\right)+C_4e^{-\frac{\sqrt{2\delta_c-1}}{2\delta_c}x}\sin\left( \frac{\sqrt{2\delta_c+1}}{2\delta_c}x\right) \end{split} \label{solution3case} \end{equation}

标签: 常微分方程, 泊松-费米方程

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