New solitary wave and other exact solutions of the van der Waals normal form for granular materials

Zafar, Asim; Qureshi, Tahir Mushtaq; Malik, Aslam; Bekir, Ahmet

doi:10.1016/j.joes.2021.07.009

Cited by 14 publications

(9 citation statements)

References 29 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%

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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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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%

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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“…In this research work, we solve the general Riccati equation through several different function transformation and obtain many new types of hyperbolic function solutions, which greatly extend the earlier Riccati equation method [16]. Then, we use this general Riccati equation as an auxiliary equation to solve the Van der Waals normal form [17][18][19][20] and obtain many new types of solitary wave interaction solutions. On the one hand, this method greatly simplifies the process of solving the Van der Waals normal form.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2022

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In this paper, we use general Riccati equation to construct new solitary wave solutions of the Van der Waals normal form, which is one of the most famous models for natural and industrial granular materials. It is very important to understand the static and dynamic characteristics of this models in many application fields. We solve the general Riccati equation through different function transformation, and many new hyperbolic function solutions are obtained. Then, it is substituted into the Van der Waals normal form as an auxiliary equation. Abundant types of solitary-wave solutions are obtained by choosing different coefficient in the general Riccati equation, and some of them have not been found in other documents. The results show that the analysis method we used is very simple and effective for dealing with nonlinear models.

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“…Over the past few years, several efficient analytical techniques for NLEEs have been suggested as the residual power series method [13], Kudryashov's method [14], stability analysis [15], extended

\left({G}&amp;#x0005E;{\prime }/G\right)

‐expansion method [16], generalized

\left({G}&amp;#x0005E;{\prime }/G\right)

‐expansion method [17], Sardar sub‐equation method [18], advanced exp

\left(-\varphi \left(\xi \right)\right)

‐expansion method [19], modified extended tanh‐function method [20], modified Khater method [21], Lie symmetry analysis [22, 23], conservation laws [24], improved Hirota bilinear method [25], generalized exponential rational function method [26, 27], rational sine‐Gordon expansion method [28], extended sinh‐Gordon equation expansion method [29], Adomian decomposition method [30], solitary wave anzatze method [31],

{\exp}_{\alpha }

function method [32], new extended direct algebraic method (NEDAM) [33], extended simple equation method (ESEM) [34], direct algebraic method [35], two‐variable

\left({G}&amp;#x0005E;{\prime }/G;1/G\right)

‐expansion approach [36], solitary wave solutions [37], exponential rational function method [38], auxiliary equation mapping method [39],

false(G^{'} false/ G

…”

Section: Introductionmentioning

confidence: 99%

Exploration of interactional phenomena and multi‐wave solutions of the fractional‐order dual‐mode nonlinear Schrödinger equation

Kopçasız,

Yaşar

2023

Math Methods in App Sciences

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This work concerns the fractional‐order dual‐mode nonlinear Schrödinger equation (FDMNLSE), which portrays the augmentation or absorption of dual waves. This model dissects the concurrent generation of two characteristic waves dealing with dual modes and introduces three physical parameters: nonlinearity, phase velocity, and dispersive factor. In the context of photonics, NLSE models the propagation of soliton pulses over intercontinental distances. Throughout this work, the fractional derivative is given in terms of time and space conformable sense. We analyze the multi‐waves method, homoclinic breather approach, and interactional solution with the double ‐functions procedure, and their applications for this equation are obtained using logarithmic transformation. The multi‐wave method is a well‐known phenomenon in nonlinear science that describes the interaction of three waves that satisfy certain resonance conditions. A breather wave is a localized and oscillatory solution that maintains its shape over time. Finally, we will discuss the dynamics of our newly obtained solutions with the help of graphs by assigning appropriate values to the parameters. The proposed methods are straight and aggressive, so the approved form can be extended for more nonlinear models. The findings are exceptional in comparison to previous findings in the literature. These outcomes may have significance for additional investigation of such frameworks to handle the nonlinear issues in applied sciences. The obtained results help us understand fluid propagation and incompressible fluids.

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New solitary wave and other exact solutions of the van der Waals normal form for granular materials

Cited by 14 publications

References 29 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 Solitary-Wave Solutions of the Van der Waals Normal Form for Granular Materials via New Auxiliary Equation Method

Exploration of interactional phenomena and multi‐wave solutions of the fractional‐order dual‐mode nonlinear Schrödinger equation

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