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Abundant accurate analytical and semi-analytical solutions of the positive Gardner–Kadomtsev–Petviashvili equation

Abstract This research article examines the correctness of two new analytical methods for solving the internal solitary waves of shallow seas. To get the computational solutions to the positive (2 + 1)-dimensional Gardner–Kadomtsev–Petviashvili model, the extended simplest equation and modified Kudryashov methods are used. Numerous new traveling wave solutions in various forms are developed in order to assess the starting conditions required for the variational iteration technique, one of the most accurate semi-analytical methods. The semi-analytical solutions are used to demonstrate the precision of the solutions obtained and the analytical methods employed. The dynamical behavior of internal solitary waves in shallow waters is shown using many three-dimensional drawings. The performance of the used schemes demonstrates their efficacy and power, as well as their capacity to handle a large number of nonlinear evolution equations.