The interference wave and the bright and dark soliton for two integro-differential equation by using BNNM

Zhang, Run-Fa; Li, Mingchu; Cherraf, Amina; Vadyala, Shashank Reddy

doi:10.1007/s11071-023-08257-5

Cited by 89 publications

(5 citation statements)

References 36 publications

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“…Substituting the obtained coefficient solution equation (21) into equation (20), and consider the transformation of equation ( 5), the following analytical solutions are obtained: In equation (22), we choose the following parameters:…”

Section: Bright and Dark Soliton Solutionmentioning

confidence: 99%

“…Zhang introduced the neural network structure into the bilinear method and proposed a new method, namely the bilinear neural network method (BNNM) [18]. By applying BNNM, one can obtain numerous novel and intriguing exact solutions [19][20][21]. Compared to existing classical methods, BNNM is able to obtain error-free exact analytical solutions of the equations by constructing some single layer or deep layer activation functions, instead of numerical solutions with errors.…”

Section: Introductionmentioning

confidence: 99%

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Lump waves, bright-dark solitons and some novel interaction solutions in (3+1)-dimensional shallow water wave equation

Lei,

Tian,

2024

Phys. Scr.

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The (3+1)-dimensional Geng equation is an extended version of the KdV model that describes the wave dynamics behavior of shallow water waves in complex applications. In this study, we discuss the (3+1) dimensional Geng equation using the bilinear neural network method. By incorporating specific activation functions into the neural network model, new test functions are constructed. Using symbolic computational techniques and selecting appropriate parameters, we systematically obtain new meaningful exact solutions of some (3+1)-dimensional Geng equations, including dark lump solutions, three kinds of interaction solutions, and bright and dark soliton solutions. Furthermore, the results are visualized through diagrams of different categories, which intuitively demonstrate the evolution process and physical characteristics of the waves.

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Section: Bright and Dark Soliton Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lump waves, bright-dark solitons and some novel interaction solutions in (3+1)-dimensional shallow water wave equation

Lei,

Tian,

2024

Phys. Scr.

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“…However, in recent years, the rapid development in the field of deep learning has provided new perspectives for solving NLPDEs.Zhou et al obtained bright solitons, breathers, and rogue waves for the third-order nonlinear Schrödinger equation by using physically informed neural networks (PINNs) [9]. Meanwhile, based on the bilinear methods and neural network models, Zhang et al proposed the bilinear neural network method (BNNM) for solving partial differential equations [10][11]. Moreover, different from the physically informative neural networks (PINNs) method [12], this method can obtain the exact analytical solutions of nonlinear partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

A fourth-order nonlinear equation studied by using amultivariate bilinear neural network method

Zhang,

Liu

2024

Preprint

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In this work, a more accurate analytical solution of nonlinear par-tial differential equation is sought by setting the generalized activation func-tion in the model of multiple bilinear neural network method. As an example,the 3-2-2-1, 3-2-3-1, 3-3-2-1 and 3-3-3-1 models are selected to study the new(2+1) dimensional nonlinear wave equation equations. Exact analytical so-lutions with arbitrary activation functions are obtained by selecting differentactivation functions and the dynamical properties are demonstrated throughthree-dimensional, two-dimensional and density plots. 2010 Mathematics Subject Classification: 35Q51, 35G99, 33F10

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“…During which various methods were proposed, for instance the Hirota bilinear transformation [5], the homologous analysis method [6], tanh-function method [7], the differential transformation method [8], and the bilinear neural network method(BNNM) [9,10] etc. Many scholars have obtained many types of analytical solutions for NLPDEs through various methods, including lump solution [11][12][13], breath solution [14][15][16], interference wave solution [17], soliton solution [18][19][20], interaction solution [21][22][23], and so on.…”

Section: Introductionmentioning

confidence: 99%

Lump solution, lump and soliton interaction solution, breathing solution, and interference wave solution for the (3+1)-dimensional fourth-order nonlinear equation by bilinear neural network method

Chen,

Ma,

Liu

et al. 2024

Preprint

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In this article, the bilinear neural network method(BNNM) is used for the first time to investigate the (3+1)-dimensional fourth-order nonlinear equation. Various bilinear neural network models are created by building diverse layers of neural networks. The precise analytical solutions of this equation consists of lump solution, lump and soliton interaction solution, breathing solution, and interference wave solution. To better reveal the dynamic behavior and physical characteristics of these solutions, constructed 3D maps, contour maps, and density maps. The method adopted in this article helps study nonlinear partial differential equations in nonlinear dynamics, oceanography, and fluid mechanics. Finally, we hope these solutions we obtained can explain some nonlinear phenomena of high-order equations.

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The interference wave and the bright and dark soliton for two integro-differential equation by using BNNM

Cited by 89 publications

References 36 publications

Lump waves, bright-dark solitons and some novel interaction solutions in (3+1)-dimensional shallow water wave equation

Lump waves, bright-dark solitons and some novel interaction solutions in (3+1)-dimensional shallow water wave equation

A fourth-order nonlinear equation studied by using amultivariate bilinear neural network method

Lump solution, lump and soliton interaction solution, breathing solution, and interference wave solution for the (3+1)-dimensional fourth-order nonlinear equation by bilinear neural network method

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