  We extend the theory for a Josephson junction travelling wave parametric
amplifier (TWPA) operating in the three-wave mixing regime and we propose a
scheme for achieving high gain. The continuous three-mode model [P. K. Tien, J.
Appl. Phys. 29, 1347 (1958)] is on one hand extended to describe a discrete
chain of Josephson junctions at high frequencies close to the spectral cutoff
where there is no up-conversion. On the other hand, we also develop a
continuous multimode theory for the low-frequency region where the frequency
dispersion is close to linear. We find that in both cases the gain is
significantly reduced compared to the prediction by the continuous three-mode
model as the result of increasingly strong dispersion at the high frequencies
and generation of up-converted modes at the low frequencies. The developed
theory is in quantitative agreement with experimental observations. To recover
the high gain, we propose to engineer a chain with dispersive features to form
a two-band frequency spectrum and to place the pump frequency within the upper
band close to the spectral cutoff. We prove that there exists a sweet spot,
where the signal and the pump are phase matched, while the up-conversion is
inhibited. This results in a high gain which grows exponentially with the
length of the TWPA.
