On the integrability of the (1+1)‐dimensional and (2+1)‐dimensional Ito equations

Wang, Yun-Hu

doi:10.1002/mma.3056

Cited by 11 publications

(4 citation statements)

References 30 publications

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“…By employing the extended homoclinic test and bilinear method, the exact soliton solutions have been studied [36]. Based on Hirota bilinear method and the Bell polynomials method, the bilinear representations, Lax pairs and N-soliton solutions have been constructed by Wang [37]. The main purpose of this paper is to construct novel exact analytical solutions to equation (9) based on the bilinear neural network model and related tensor formula.…”

Section: Novel Exact Solutions To the (2 + 1)-dimensional Ito Equationmentioning

confidence: 99%

Diversity of exact solutions to the (2+1)-dimensional Ito equation via bilinear neural network method

Ma,

Sudao

2023

Phys. Scr.

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Recently, searching for exact solutions to nonlinear partial differential equations has gradually become a hot research topic. It is of great scientific research and application value to reveal the law of wave propagation, explain natural phenomena accurately and apply related technologies scientifically. In this paper, bilinear neural network method (BNNM) was employed to obtain some new exact analytical solutions to the (2 + 1)-dimensional Ito equation. Based on the Hirota form of Ito equation, we constructed activation functions f in various forms containing the exp(ξ), sin(ξ), cos(ξ), cosh(ξ) and squares of polynomial functions in multi-layer neurons such as [3-2-2] and [3-2-3] neural network models. The test function f in this work is a new expression. On the other hand, these solutions have not been studied yet. As a result, we obtained several new interaction solutions, such as periodic wave solution, breather solution and bright-dark solitons solution, etc. In addition, the corresponding 3D, density and contour plots of all the solution forms were drawn and their characters and dynamic behaviors were vividly demonstrated.

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Section: Novel Exact Solutions To the (2 + 1)-dimensional Ito Equationmentioning

confidence: 99%

Diversity of exact solutions to the (2+1)-dimensional Ito equation via bilinear neural network method

Ma,

Sudao

2023

Phys. Scr.

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“…Moreover, the existence of infinite conservation laws indicates the integrability of a given NLEE. Starting from the Bell polynomials (20), we will derive the infinite conservation laws of equation (2).…”

Section: Infinite Conservation Lawsmentioning

confidence: 99%

“…Based on the close relations between the Bell polynomials and Hirota D-operators, Lambert and his coworkers have proposed a direct and systematic method, which can be used to derive integrable structures such as bilinear representation, bilinear transformation (BT), and Lax pairs [13,14]. The Bell polynomials approach has been applied to investigate a large number of NLEEs and also has been extended to the variable-coefficient, supersymmetric, and discrete soliton equations [15][16][17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

New bilinearization, Bäcklund transformation and infinite conservation laws for the KdV6 equation with Bell polynomials

Wazwaz

2015

Math Methods in App Sciences

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By using the Bell polynomials method and symbolic computation, we study the integrability of the KdV6 equation. We develop, in this work, new results regarding the integrability concept. We show that the newly developed bilinear representation and bilinear Bäcklund transformation are different from those reported in the literature. Moreover, we firstly present the infinite conservation laws, and the conserved densities and fluxes are given in explicit recursion formulas.

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“…Furthermore, the integrability of Eq. ( 1) was proven by the Lax pair through the Bell polynomial technique [19]. The lump solutions and the interaction solutions between lumps and line solitons were obtained by the direct method, which was based on the Hirota bilinear form involving a combination of quadratic and other types of functions [20]- [23].…”

Section: Introductionmentioning

confidence: 99%

Integrability and exact solutions of the (2+1)-dimensional variable coefficient Ito equation

Chu,

Liu,

Chen

2023

Nonlinear Dyn

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This paper focuses on the study of the (2+1)-dimensional Ito equation with spatially variable coefficients. The Lax integrability of the equation is confirmed by the Lax pair derived from the Bell polynomials. In addition, the Hirota bilinear form, the Bäcklund transformation and infinite conservation laws of the equation are obtained by the Bell polynomials. More importantly, the ansatz method and the Hirota bilinear method are employed to construct different types of N-soliton solutions and interaction solutions for the (2+1)-dimensional Ito equation with different coefficient functions. All these solutions are presented graphically.

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On the integrability of the (1+1)‐dimensional and (2+1)‐dimensional Ito equations

Cited by 11 publications

References 30 publications

Diversity of exact solutions to the (2+1)-dimensional Ito equation via bilinear neural network method

Diversity of exact solutions to the (2+1)-dimensional Ito equation via bilinear neural network method

New bilinearization, Bäcklund transformation and infinite conservation laws for the KdV6 equation with Bell polynomials

Integrability and exact solutions of the (2+1)-dimensional variable coefficient Ito equation

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