Novel analytical cnoidal and solitary wave solutions of the Extended Kawahara equation

El-Tantawy, S. A.; Salas, Alvaro H.; Alharthi, M.R.

doi:10.1016/j.chaos.2021.110965

Cited by 49 publications

(17 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Given the importance of this equation, it remains the subject of study for many researchers. Many different techniques were devoted to studying various solutions of Equation ( 1) and its family [4][5][6][7][8][9][10][11][12][13]. For example, Kudryashov [14], obtained exact meromorphic solutions of the Kawahara equation using the Laurent series.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

Tian

Zhang

et al. 2022

Axioms

View full text Add to dashboard Cite

In this article, we prove that the ⟨p,q⟩ condition holds, first by using the Fuchs index of the complex Kawahara equation, and then proving that all meromorphic solutions of complex Kawahara equations belong to the class W. Moreover, the complex method is employed to get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Kawahara equation. Our results reveal that all rational solutions ur(x+νt) and simply periodic solutions us,1(x+νt) of Kawahara equation are solitary wave solutions, while simply periodic solutions us,2(x+νt) are not real-valued. Finally, computer simulations are given to demonstrate the main results of this paper. At the same time, we believe that this method is a very effective and powerful method of looking for exact solutions to the mathematical physics equations, and the search process is simpler than other methods.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

Tian

Zhang

et al. 2022

Axioms

View full text Add to dashboard Cite

show abstract

“…There is a large group of fluid mechanics and plasma physicists researchers who have worked a lot on this equation and many related equations in multidimensional (Kadomtsev-Petviashvili (KP) equation, Zakharov-Kuznetsov (ZK) equation, etc.) to investigate the propagation of many nonlinear structures in different models of plasmas [8][9][10][11][12]. It is known that solitary waves can be created and propagated in any system if the balance between the wave dispersion and nonlinearity is fulfilled.…”

Section: Introductionmentioning

confidence: 99%

New Localized and Periodic Solutions to a Korteweg–de Vries Equation with Power Law Nonlinearity: Applications to Some Plasma Models

2022

Self Cite

View full text Add to dashboard Cite

Traveling wave solutions, including localized and periodic structures (e.g., solitary waves, cnoidal waves, and periodic waves), to a symmetry Korteweg–de Vries equation (KdV) with integer and rational power law nonlinearity are reported using several approaches. In the case of the localized wave solutions, i.e., solitary waves, to the evolution equation, two different methods are devoted for this purpose. In the first one, new hypotheses with Cole–Hopf transformation are employed to find general solitary wave solutions. In the second one, the ansatz method with hyperbolic sech algorithm are utilized to obtain a general solitary wave solution. The obtained solutions recover the solitary wave solutions to all one-dimensional KdV equations with a power law nonlinearity, such as the KdV equation with quadratic nonlinearity, the modified KdV (mKdV) equation with cubic nonlinearity, the super KdV equation with quartic nonlinearity, and so on. Furthermore, two different approaches with two different formulas for the Weierstrass elliptic functions (WSEFs) are adopted for deriving some general periodic wave solutions to the evolution equation. Additionally, in the form of Jacobi elliptic functions (JEFs), the cnoidal wave solutions to the KdV-, mKdV-, and SKdV equations are obtained. These results help many authors to understand the mystery of several nonlinear phenomena in different branches of sciences, such as plasma physics, fluid mechanics, nonlinear optics, Bose Einstein condensates, and so on.

show abstract

“…The exact result of such nonlinear phenomena may not be possible for some physical problems. For instance in a plasma physics, there are many nonintegrable PDEs that can not support exact analytic solutions such as the integer and fractional damped thirdorder KdV-type equations and the damped integer and fractional fifth-order KdV-type equations (the family of damped Kawahara equation) and many other equations related to plasma physics [35][36][37][38][39][40]. Moreover, in non-Maxwellian plasma models that have trapped particles follow nonisothermal or Schamel distribution in addition to the particle kinematic viscosity, in this case, the fluid equations of the plasma model can be reduced to a nonintegrable damped Schamel KdV-Burgers equation [41,42].…”

Section: Introductionmentioning

confidence: 99%

“…The mathematical model for exploring dispersive wave phenomena in several research areas is the KdV equations, such as quantum mechanics, fluid dynamics, optics, and plasma physics [44,45]. Fifth-order KdV/Kawahara form equations utilized to analyze different nonlinear phenomena in particle physics and in plasma physics [38][39][40]. It plays a vital function in the distribution of waves [50].…”

Section: Introductionmentioning

confidence: 99%

Study of Fuzzy Fractional Third-Order Dispersive KdV Equation in a Plasma under Atangana-Baleanu Derivative

Areshi

El-Tantawy

Alotaibi

et al. 2022

Journal of Function Spaces

View full text Add to dashboard Cite

Motivated by the wide-spread of both integer and fractional third-order dispersive Korteweg-de Vries (KdV) equations in explaining many nonlinear phenomena in a plasma and many other fluid models, thus, in this article, we constructed a system for calculating an analytical solution to a fractional fuzzy third-order dispersive KdV problems. We implemented the Shehu transformation and the iterative transformation technique under the Atangana-Baleanu fractional derivative. The achieved series result was contacted and determined the analytic value of the suggested models. For the confirmation of our system, three various problems have been represented, and the fuzzy type solution was determined. The fuzzy results of upper and lower section of all three problems are simulate applying two different fractional orders among zero and one. Because it globalises the dynamic properties of the specified equation, it delivers all forms of fuzzy solutions occurring at any fractional order among zero and one. The present results can help many researchers to explain the nonlinear phenomena that can create and propagate in several plasma models.

show abstract

Novel analytical cnoidal and solitary wave solutions of the Extended Kawahara equation

Cited by 49 publications

References 34 publications

All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

New Localized and Periodic Solutions to a Korteweg–de Vries Equation with Power Law Nonlinearity: Applications to Some Plasma Models

Study of Fuzzy Fractional Third-Order Dispersive KdV Equation in a Plasma under Atangana-Baleanu Derivative

Contact Info

Product

Resources

About