On a coupled Kadomtsev–Petviashvili system associated with an elliptic curve

The coupled Kadomtsev–Petviashvili system associated with an elliptic curve, proposed by Date, Jimbo, and Miwa [J. Phys. Soc. Jpn., 52:766–771, 1983], is reinvestigated within the direct linearization framework, which provides us with more insights into the integrability of this elliptic model from the perspective of a general linear integral equation. As a result, we successfully construct for the elliptic coupled Kadomtsev–Petviashvili system not only a Lax pair composed of differential operators in 2 × 2 matrix form but also multisoliton solutions with phases parametrized by points on the elliptic curve. Dimensional reductions based on the direct linearization, to the elliptic coupled Korteweg–de Vries and Boussinesq systems, are also discussed. In addition, a novel class of solutions is obtained for the (Formula presented.) -type Kadomtsev–Petviashvili equation with nonzero constant background as a by-product.