Patterns of Lumps for the Variant Differential-Difference Kadomtsev-Petviashvili Equation Via Integer Partitions

This paper investigates high-order lump patterns in a novel differential-difference KP equation, derived through the introduction of a new class of trigonometric-type bilinear Hirota operators. Rational solutions are obtained by applying two differential operators to the elements of Gram-type determinants, and are succinctly expressed in terms of Schur polynomials, establishing a direct connection between the lump patterns and Schur functions. Using concepts from integer partitions, we systematically construct these high-order patterns. Furthermore, asymptotic analysis in the large-parameter regime reveals that the distribution of lump centers is analytically governed by the root structures of special polynomials, including the Yablonskii-Vorob’ev, Umemura, Wronskian-Hermite, and Okamoto polynomials.