Abstract:
This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with timedependent variable coefficients that describe the nonlinear wave propagation of small-amplitude
surface waves in narrow channels or large straits with slowly varying width and depth and nonvanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in
(2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique,
we carry out infinitesimal generators, potential vector fields, and various similarity reductions of
the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via
two similarity reductions. The closed-form analytic solutions are achieved, including in the shape
of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped
soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave
solitons. All the obtained solutions are verified and validated by using back substitution to the
original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these
obtained solutions with some three-dimensional graphics via numerical simulation. The obtained
variable coefficient solutions are more relevant and useful for understanding the dynamical structures
of nonlinear KP equations and shallow water wave models.
