\begin{equation}
\begin{split}
\begin{aligned}
\frac{\partial v_{i,n}}{\partial t}+\lambda_{3-i}\frac{\partial v_{i,n}}{\partial x}=\frac{1}{\sqrt{\Delta}}\left(\frac{\partial\lambda_{i}}{\partial t}+\lambda_{3-i}\frac{\partial\lambda_{i}}{\partial x}\right)\left(v_{2,n}-v_{1,n}\right)\\
&+\tilde{f}_{n}
\end{aligned}
\end{split}
\end{equation}
