Abstract
In the current study, an analytical treatment is studied starting from the 2 + 1 -dimensional generalized Hirota-Satsuma-Ito (HSI) equation. Based on the equation, we first establish the evolution equation and obtain rational function solutions by means of the bilinear form with the help of the Hirota bilinear operator. Then, by the suggested method, the periodic, cross-kink wave solutions are also obtained. Also, the semi-inverse variational principle (SIVP) will be utilized for the generalized HSI equation. Two major cases were investigated from two different techniques. Moreover, the improved tan χ ξ method on the generalized Hirota-Satsuma-Ito equation is probed. The 3D, density, and contour graphs illustrating some instances of got solutions have been demonstrated from a selection of some suitable parameters. The existing conditions are handled to discuss the available got solutions. The current work is extensively utilized to report plenty of attractive physical phenomena in the areas of shallow water waves and so on.
Highlights
Nonlinear partial differential equations containing time variables are generally referred to as nonlinear evolution equations (NLEEs) which can describe the state or process changing along with times in physics, dynamics, and other nature sciences
We consider the ð3 + 1Þ-dimensional generalized Hirota-Satsuma-Ito (HSI) shallow water wave equation which will be read [28] as Advances in Mathematical Physics
Where ζjð0 ≤ j ≤ σÞ, θjð0 ≤ j ≤ σÞ are constants to be determined, such that ζσ, θσ ≠ 0, and χ = χðξÞ is the solution of the following first order differential equation: χ′ = α1 sin ðχÞ + α2 cos ðχÞ + α3: ð82Þ
Summary
Nonlinear partial differential equations containing time variables are generally referred to as nonlinear evolution equations (NLEEs) which can describe the state or process changing along with times in physics, dynamics, and other nature sciences. Liu et al [32] investigated the N-soliton solution to construct the ð2 + 1Þ -dimensional generalized Hirota-Satsuma-Ito equation, from which some localized waves such as line solitons, lumps, periodic solitons, and their interactions. By using Hirota’s bilinear form and the extended Ansatz function method, Liu and Ye got the new exact periodic cross-kink wave solutions for the ð2 + 1Þ-dimensional KdV equation [38]. A few conclusions and outlook will be given in the final section
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