Cauchy matrix approach to the nonisospectral and variable-coefficient Kadomtsev–Petviashvili equation

Cauchy matrix approach is developed to construct the nonisospectral and variable-coefficient equations and study their integrability. We derive the nonisospectral and variable-coefficient Kadomtsev–Petviashvili (n-vcKP) equation, which includes the standard KP equation and the nonisospectral and variable-coefficient KdV equation as special cases. The connection of the $\tau$ function of the n-vcKP equation with the Cauchy matrix approach is clarified. The Lax pair for the n-vcKP equation is derived in a systematic way. Two types of exact solutions are found by solving the corresponding Sylvester equation.