topological order - FQH Edge Theory as decoupled chiral bosons

The action describing the edge theory of the Fractional Quantum Hall effect is given by

$$\begin{equation} S = \frac{1}{4\pi} \int \mathrm{d}x \ \mathrm{d}t \left[ K_{IJ} \ \partial_{t}\phi_{RI} \partial_{x}\phi_{RJ} - V_{IJ} \partial_{x}\phi_{RI}\partial_{x}\phi_{RJ} \right] \end{equation}$$ for scalar fields $\phi_{RI}$ with $I=1....dim(K)$ and some symmetric,invertible matrix K and some positive definite, symmetric "velocity matrix" V.

The equations of motion for the fields read

$$\begin{equation} \partial_{t}\partial_{x} K_{IJ}\phi_{RJ} - \partial_{x}^{2}V_{IJ}\phi_{RJ} = 0. \end{equation}$$ This smells already a lot like the chiral boson theory in 1+1D Minkowski space which this action is supposed to be equivalent to. However I do not see the transformation that "decouples" the fields. Can anyone help?

Thanks in advance for your responses!