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Algebraic curves have significant applications in the study of the Kadomtsev-Petviashvili hierarchy, which is a universal integrable system that describes shallow water waves. The corresponding finite-genus solutions are expressed in terms of the Riemann theta function. The theta function degenerates as the curve becomes more singular, leading to soliton and rational solutions. These solutions, in turn, characterize the curves in the moduli space of abelian varieties, thus providing a solution to the Schottky problem. To better understand the bilateral connection, we explore various perspectives in computational algebraic geometry, including transcendental and combinatorial approaches, while also utilizing mathematical software. Our study yields new results and insights, along with future directions for research in this field.