Newly exploring the Lax pair, bilinear form, bilinear Bäcklund transformation through binary Bell polynomials, and analytical solutions for the (2 + 1)-dimensional generalized Hirota–Satsuma–Ito equation

Abstract

The (2+1)-dimensional generalized Hirota–Satsuma–Ito equation describing the numerous wave dynamics in shallow waters is investigated in this study. The integrable characteristics of the aforesaid equation, such as a bilinear Bäcklund transformation and Lax pair, are revealed using the Bell polynomials method. First, using this technique, with the aid of Hirota operators, the bilinear form is constructed for the considered equation. In addition, the bilinear Bäcklund transformation and the Lax pair of the aforesaid equation are derived successfully using the bilinear form. Moreover, the bilinear form is also used to construct analytical solutions utilizing the three-wave approach with a test function. While using this method, numerous analytical solutions are derived, which are not presented in the literature. A three-dimensional graph has been plotted for each of the obtained results by giving the appropriate values of the free parameters. These plots reveal a wide variety of wave behavior, such as kink-soliton, periodic wave, anti-kink soliton, and complex periodic wave solutions.

Abundant Closed-Form Soliton Solutions to the Fractional Stochastic Kraenkel–Manna–Merle System with Bifurcation, Chaotic, Sensitivity, and Modulation Instability Analysis

“…For deeper consideration of nonlinear occurrence and realistic challenges, it is essential to discover closed-form soliton solutions of SFPDEs. According to the quick advancements in nonlinear sciences, a variety of simple and efficient approaches have been developed to obtain closed-form soliton solutions to NLPDEs, including the Hirota method [4,5], the Bernoulli sub-equation method [6,7], the F-expansion method [8], the (G ′ /G 2 )-expansion method [9], the simple equation method [10], the modified auxiliary equation method [11,12], the two variable (G ′ /G, 1/G)-expansion method [13][14][15][16], the Lie symmetric analysis [17], the polynomial complete discriminant system [18], the tanh-coth scheme [19], the Conservation laws method [20], the generalized exponential rational function approach [21,22], the binary bell polynomials method [23], the mapping method [24], the Shehu transform scheme [25], the sine-Gordon expansion [26], the Cole-Hopf transformation method [27,28], the Fan subequation technique [29], the unified method [30], the Khater method [31], the r + mEDAM method [32], the spectral Tau method [33], the G ′ G ′ +G+A -expansion procedure [34][35][36][37][38], the sub-equation method [39], the collocation method [40], the finite element method [41], and the generalized G ′ /G-expansion method …”

Extended (3 + 1)-dimensional Kairat-II and Kairat-X equations: Painlevé integrability, multiple soliton solutions, lump solutions, and breather wave solutions

“…The two constructed equations give a good sign of developing new integral equations away from the Korteweg–de Vries equation or other well-known systems. Integrable properties like bilinear transformation, Lax pairs, Bäcklund bilinear transformation, Hirota’s bilinear method and infinite conservation laws have been extensively used to obtain the exact solutions to these equations (Singh and Saha Ray, 2023a, 2023b, 2024). Moreover, integrability is the key step to ensuring the analytic solvability of nonlinear equations (Singh and Saha Ray, 2023b).…”

Multi-lump, resonant Y-shape soliton, complex multi kink solitons and the solitary wave solutions to the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for incompressible fluid

“…As is well known, a series of important principles such as the linear superposition principle of solutions no longer hold for NPDEs, so there is no universal method for solving the NPDEs. Although there is no universal and effective method for obtaining the exact solutions to NPDEs, several approaches for constructing the exact solutions for the NPDEs have been put forward in different applicable situations such as the extended F-expansion technique [1][2][3][4], Darboux transformation approach [5][6][7], general integral technique [8], unified Riccati equation approach [9], Bäcklund transformation [10][11][12][13], exp-function method [14][15][16][17], Subequation technique [18][19][20], tanh function method [21,22], (G′/G)-expansion approach [23,24] and many others [25][26][27][28][29]. In this exploration, we aim to give a study to the (2+1)-dimensional BLMPE, which reads as:…”