Jacobi Elliptic Function Solutions and Traveling Wave Solutions of the (2 + 1)‐Dimensional Gardner‐KP Equation

Abstract

The fully integrable KP equation is one of the models that describes the evolution of nonlinear waves, the expansion of the well‐known KdV equation, where the impacts of surface tension and viscosity are negligible. This paper uses the Modified Extended Direct Algebraic (MEDA) method to build fresh exact, periodic, trigonometric, hyperbolic, rational, triangular and soliton alternatives for the (2 + 1)‐dimensional Gardner KP equation. These solutions that we discover in this article will help us understand the phenomena of the (2 + 1)‐dimensional Gardner KP equation. Comparing the study in this paper and existing work, we find more exact solutions with soliton and periodic structures and the rational function solution in this paper is more general than the rational solution in existing literature. Most of the Jacobi elliptic function solutions and the mixed Jacobi elliptic function solutions to the (2 + 1)‐dimensional Gardner KP equation discovered in this paper, to the best of our highest understanding are not seen in any existing paper until now.

Optimal Subalgebra of GKP by Using Killing Form, Conservation Law and Some More Solutions

“…( 25)-( 27), ( 34)-(38), and (41)-(43) via using Lie-group theory. These solutions are different from previous findings [1][2][3][4][5][6][7][8][9][10][11][12] and involve arbitrary constants C 1 , C 2 , c 1 , c 2 , α and β etc. We have used symbolic computations in MATLAB to generate animations.…”

“…9, 10, 11, only spatial variation (−100 ≤ x ≤ 100) is taken. Previous researches [1][2][3][4][5][6][7][8][9][10][11][12] are related to solutions of GKP which are obtained using different techniques/issues. Here, a frame of animation is shown in each Fig., which dominantly performs the wave profile during the whole span of time.…”

“…For instance, among the past researches [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], exact solutions of NPDEs play a vital role in the study of these complex physical phenomena. This literature review includes the previous findings [1][2][3][4][5][6][7][8][9][10][11][12] during the last decade. We have focused especially on the analytical solutions which are obtained for (2+1)-dimensional GKP.…”

“…() obtained some new analytical solution of GKP (1). Different forms of the GKP are solved by employing several methods/tools, notably (G /G, 1/G)-Expansion [1,8], Modified Extended Direct Algebraic [2], Auxiliary equation [3], Bifurcation theory [4], (G /G)-Expansion [6], Improved G /G-expansion, tanh and coth [7], Lie-symmetries and first integral method [9], Bifurcation and phase portrait [10], and some others. Different forms of the solutions of the GKP are depicted in Table 1 using these techniques.…”

“…Motivated by the above review of literature on GKP, we have generated a one-dimensional optimal subalgebra for Eq. ( 1) by using Killing form, then STM is employed to bridge the gap in previous research [1][2][3][4][5][6][7][8][9][10][11][12]. The STM was developed by Lie via one-parameter Lie-group…”