A study on analytical solutions of one of the important shallow water wave equations and its stability analysis

Abstract The present research uses the modified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:math> -expansion technique to give analytical solutions for the nonlinear time fractional integro-differential Kadomtsev-Petviashvili (KP) hierarchy equation with beta fractional derivative. The modified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:math> -expansion technique is an important method for finding analytical solutions to nonlinear equations. The suggested approach produces three sorts of solutions: trigonometric, hyperbolic, and rational. These answers have been discovered with the aid of a software tool. Additionally, stability analysis of observed governing model solutions is used to validate scientific calculations. Furthermore, the 3-dimensional, contour, and 2-dimensional images are provided for a better understanding of the behavior of the solutions.