In your opinion, how is Solomon able to keep any glimmer of hope of his freedom alive? How does he not succumb to his current fate, stopping any and all attempts at regaining his freedom?

His identity — at least outwardly — is forced to completely change. Internally, however, Northup remembers his past identity, which inspires him to survive. He finds value in the violin; he never gives up trying to escape; he is faithful to his wife; he always remembers who he is and what he wants to return to.