  Recent advancements in quantum technologies and advanced detection
experiments have underscored the pressing need for the detection of exceedingly
weak signals within the microwave frequency spectrum. Addressing this
challenge, the Josephson Traveling Wave Parametric Amplifier (JTWPA) has been
proposed as a cryogenic front-end amplifier capable of approaching the quantum
noise limit while providing a relevant bandwidth. This research is centered on
a comprehensive numerical investigation of the JTWPA, without resorting to
simplifications regarding the nonlinearity of the essential components.
Specifically, this study focuses on a thorough examination of the system,
characterized by coupled nonlinear differential equations representing all
components of the device. Proper input and output signals at the device's
boundaries are considered. The analysis of the output signals undergoing the
parametric amplification process involves a detailed exploration of phase-space
dynamics and Fourier spectral analysis of the output voltage. This study is
conducted while considering the parameters ruling the response of the device
under pump and signal excitations. In addition to the expected signal
amplification, the findings reveal that the nonlinear nature of the system can
give rise to unforeseen phenomena, depending on the system's operational
conditions, which include: the generation of pump tone harmonics, modulation of
the signal gain, and incommensurate frequency generation-effects that are not
easily accommodated by simplistic linearized approaches
