Recent research has demonstrated Reservoir Computing's capability to model various chaotic dynamical systems, yet its application to Hamiltonian systems remains relatively unexplored. This paper investigates the effectiveness of Reservoir Computing in capturing rogue wave dynamics from the nonlinear Schr\"{o}dinger equation, a challenging Hamiltonian system with modulation instability. The model-free approach learns from breather simulations with five unstable modes. A properly tuned parallel Echo State Network can predict dynamics from two distinct testing datasets. The first set is a continuation of the training data, whereas the second set involves a higher-order breather. An investigation of the one-step prediction capability shows remarkable agreement between the testing data and the models. Furthermore, we show that the trained reservoir can predict the propagation of rogue waves over a relatively long prediction horizon, despite facing unseen dynamics. Finally, we introduce a method to significantly improve the Reservoir Computing prediction in autonomous mode, enhancing its long-term forecasting ability. These results advance the application of Reservoir Computing to spatio-temporal Hamiltonian systems and highlight the critical importance of phase space coverage in the design of training data.

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