New Solitary-Wave Solutions of the Van der Waals Normal Form for Granular Materials via New Auxiliary Equation Method

Zhu, Xiaomeng; Cheng, Jinkang; Chen, Zhuokai; Wu, Guojiang

doi:10.3390/math10152560

Cited by 7 publications

(8 citation statements)

References 20 publications

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“…Compared with other published literature studies [17,[41][42][43], the solutions of Equation ( 22) obtained are novel. In these infinite groups of solutions, except for a few low-n-order solutions that have been reported in other literature studies, the rest are all our newly discovered solutions.…”

Section: Construction Of Infinite-series Solitary Wave Solutionsmentioning

confidence: 95%

“…To simplify the process as much as possible, we choose the following simple Riccati equation [17,[41][42][43]:…”

Section: Construction Of Infinite-series Solitary Wave Solutionsmentioning

confidence: 99%

“…where r, k 0 , and k 1 are constants to be determined; and n = 0, 1, 2, • • • . By substituting Equation (23) into Equation (22), using Equation (17), and then setting each coefficient of g n i (ξ) (i = 0, 1, 2, • • • ) as zero, the following equations can be obtained:…”

Section: Construction Of Infinite-series Solitary Wave Solutionsmentioning

confidence: 99%

“…where r, k 0 , and k 1 are constants to be determined; and n = 0, 1, 2, • • • . We substitute Equation (29) into Equation ( 22) and use Equation (17). The following equations can be obtained:…”

Section: Construction Of Infinite-series Solitary Wave Solutionsmentioning

confidence: 99%

“…[12,13], the auxiliary equation method in Refs. [14][15][16][17][18], the (G /G)-expansion method and its extension in Refs. [19][20][21][22], and so on.…”

Section: Introductionmentioning

confidence: 99%

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Construction of New Infinite-Series Exact Solitary Wave Solutions and Its Application to the Korteweg–De Vries Equation

Guo

2023

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The Korteweg–de Vries (KDV) equation is one of the most well-known models in nonlinear physics, such as fluid physics, plasma, and ocean engineering. It is very important to obtain the exact solutions of this model in the process of studying these topics. In the present paper, using distinct function iteration relations in two ways, namely, squaring infinitely and extracting the square root infinitely, which have not been reported in other documents, we construct abundant types of new infinite-series exact solitary wave solutions using the auxiliary equation method. Most of these solutions have not been reported in previous papers. The numerical analysis of some solutions shows complex solitary wave phenomena. Some solutions can have stable solitary wave structures, while others may have singularities in certain space–time positions. The results show that the analysis model we use is very simple and effective for the construction of new infinite-series solutions and new solitary wave structures of nonlinear models.

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Section: Construction Of Infinite-series Solitary Wave Solutionsmentioning

confidence: 95%

“…To simplify the process as much as possible, we choose the following simple Riccati equation [17,[41][42][43]:…”

Section: Construction Of Infinite-series Solitary Wave Solutionsmentioning

confidence: 99%

Section: Construction Of Infinite-series Solitary Wave Solutionsmentioning

confidence: 99%

Section: Construction Of Infinite-series Solitary Wave Solutionsmentioning

confidence: 99%

“…[12,13], the auxiliary equation method in Refs. [14][15][16][17][18], the (G /G)-expansion method and its extension in Refs. [19][20][21][22], and so on.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Construction of New Infinite-Series Exact Solitary Wave Solutions and Its Application to the Korteweg–De Vries Equation

Guo

2023

Fractal Fract

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New Complex Wave Solutions and Diverse Wave Structures of the (2+1)-Dimensional Asymmetric Nizhnik–Novikov–Veselov Equation

Guo

2023

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In this paper, we use a new, extended Jacobian elliptic function expansion method to explore the exact solutions of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (aNNV) equation, which is a nonlinear physical model to describe an incompressible fluid. Combined with the mapping method, many new types of exact Jacobian elliptic function solutions are obtained. As we use two new forms of transformation, most of the solutions obtained are not found in previous studies. To show the complex nonlinear wave phenomena, we also provide various wave structures of a group of solutions, including periodic wave and solitary wave structures of ordinary traveling wave solutions, horseshoe-type wave, s-type wave and breaker-wave structures superposed by two kinds of waves: chaotic wave structures with chaotic behavior and spiral wave structures. The results show that this method is effective and powerful and can be used to construct various exact solutions for a wide range of nonlinear models and complex nonlinear wave phenomena in mathematical and physical research.

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The Extended Direct Algebraic Method for Extracting Analytical Solitons Solutions to the Cubic Nonlinear Schrödinger Equation Involving Beta Derivatives in Space and Time

2023

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In the field of nonlinear optics, quantum mechanics, condensed matter physics, and wave propagation in rigid and other nonlinear instability phenomena, the nonlinear Schrödinger equation has significant applications. In this study, the soliton solutions of the space-time fractional cubic nonlinear Schrödinger equation with Kerr law nonlinearity are investigated using an extended direct algebraic method. The solutions are found in the form of hyperbolic, trigonometric, and rational functions. Among the established solutions, some exhibit wide spectral and typical characteristics, while others are standard. Various types of well-known solitons, including kink-shape, periodic, V-shape, and singular kink-shape solitons, have been extracted here. To gain insight into the internal formation of these phenomena, the obtained solutions have been depicted in two- and three-dimensional graphs with different parameter values. The obtained solitons can be employed to explain many complicated phenomena associated with this model.

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New Solitary-Wave Solutions of the Van der Waals Normal Form for Granular Materials via New Auxiliary Equation Method

Cited by 7 publications

References 20 publications

Construction of New Infinite-Series Exact Solitary Wave Solutions and Its Application to the Korteweg–De Vries Equation

Construction of New Infinite-Series Exact Solitary Wave Solutions and Its Application to the Korteweg–De Vries Equation

New Complex Wave Solutions and Diverse Wave Structures of the (2+1)-Dimensional Asymmetric Nizhnik–Novikov–Veselov Equation

The Extended Direct Algebraic Method for Extracting Analytical Solitons Solutions to the Cubic Nonlinear Schrödinger Equation Involving Beta Derivatives in Space and Time

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