Direct Reductions and Symmetry-generated Exact Solutions of a Generalized Potential Kadomtsev–Petviashvili Equation
Jiaxiang Qiao
School of Mathematics and Statistics, Southwest University, Chongqing, 400175, China.
Shaowei Liu *
School of Mathematics and Statistics, Southwest University, Chongqing, 400175, China and Chongqing College of Artificial Intelligence, Chongqing, 401336, China.
*Author to whom correspondence should be addressed.
Abstract
A generalised potential Kadomtsev–Petviashvili (gpKP) equation with constant mixed-derivative and longitudinal-dispersion coefficients is investigated by direct similarity reductions and explicit solution-preserving transformations. A change of independent variables shows that the two additional drift coefficients can be removed without modifying the nonlinear or transverse-dispersion terms. A travelling-wave reduction yields an autonomous second-order equation for the derivative of the potential. Rational, hyperbolic, trigonometric, and Weierstrass-zeta profiles are obtained on their respective regularity domains. A separate quadratic-transverse ansatz gives a nonautonomous reduction equivalent, for nonzero forcing, to the first Painlevé equation. An algebraic solution with a moving singular line is constructed explicitly. A time-dependent longitudinal translation, accompanied by polynomial terms, provides an infinite-dimensional family of solution-preserving transformations; constant shears and scaling transformations are also exhibited. Each reported family is supported by a substitution identity for the governing partial differential equation. The corresponding surface plots distinguish bounded line-wave profiles from meromorphic solutions with poles. No global finite-energy or nonlinear stability property is inferred from these exact expressions.
Keywords: Generalised potential Kadomtsev–Petviashvili equation, Clarkson–Kruskal reduction, exact solution, Painlevé I equation, symmetry transformation, singular wave