Flat and almost flat bands in the quasi-one-dimensional Josephson junction array

The dispersion law for the linear waves in the quasi-one-dimensional array of inductively coupled Josephson junctions (JJ) is derived. The array has a multiladder structure that consists of the finite number of rows ($N\ge 2$) in $Y$ direction and is infinite in $X$ direction. The spectrum of the linear waves (Josephson plasmons) consists of $2N-1$ branches. Among these branches there is a $N$-fold completely flat degenerate one that coincides with the Josephson plasma frequency. The remaining $N-1$ branches have a standard Josephson plasmon dispersion law typical for 1D JJ arrays. Application of the uniform dc bias on the top of each vertical column of junctions lifts the degeneracy and only one flat branch remains unchanged. The rest of the previously flat branches become weakly dispersive. The parameter range where the flatness of these branches is maximal has been discussed.