This work studies the first equation of the Kadomtsev-Petviashvili (KP) hierarchy. The sine-Gordon expansion method
(SGEM) and the rational SGEM (RSGEM) are applied to the governing model. RSGEM is the developed version of SGEM.
New complex travelling wave solutions, logarithmic and complex function properties are obtained. Several simulations
such as 2D, 3D and contour surfaces of the obtained results are plotted. Physical meanings of these solutions are also
reported. Strain conditions are also extracted.