Employing the KP reduction approach, the primary goal of this research work is to investigate the soliton and rational solutions of the \((3+1)\) -dimensional nonlocal Mel’nikov equation with non-zero background. The solutions presented are all \(N\times N\) Gram determinants. In contrast to the previous exact solutions of the nonlocal model obtained by the KP reduction method, we introduce two types of parameter constraints into the \(\tau \) function. This leads to the appearance of rational solutions and soliton (breather) solutions against a background of periodic wave. In particular, the soliton types we obtained are dark soliton, antidark soliton, breather, periodic wave and degenerate soliton. Furthermore, it has been discovered that lumps can appear in odd or even numbers in two backgrounds, which is a novel finding. The dynamic behavior of all solutions has been comprehensively analyzed.
