Cauchy Matrix Structure Underlying the BKP Equation and its Dimensional Reductions

This paper establishes a Cauchy matrix scheme for the BKP equation based on a specific Sylvester equation and corresponding master functions. Within this framework, we systematically derive the BKP equation together with its τ-function and Lax pair. Through dimensional reductions, the BKP system is reduced to the SK and bSK equations, and their exact solutions are obtained. Furthermore, starting from a soliton solution of the BKP equation, we construct both singular solutions that exhibit isolated blow-up points and regular (non-singular) solutions. Graphical illustrations of these singular and regular solutions are provided.