New traveling solutions of the fractional nonlinear KdV and ZKBBM equations with 𝒜ℬℛ fractional operator

2021 ◽  
pp. 2150232
Author(s):  
Mostafa M. A. Khater
Keyword(s):  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Interaction Model ◽  
Gravity Models ◽  
Fractional Operator ◽  
Solitary Wave Solutions ◽  
Trigonometric Functions ◽  
Wave Solutions ◽  
Simplest Equation Method ◽  
First Time

This research paper investigates novel explicit wave solutions of the fractional Korteweg–de Vries (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. These models are used as gravity models in water and an interaction model between the long waves. The Atangana–Baleanu ([Formula: see text]) fractional operator is utilized for the first time to convert the fractional form of both models into nonlinear partial differential equations with an integer order. The extended simplest equation method is employed to construct some distinct types of solitary wave solutions such as exponential, rational, hyperbolic and trigonometric functions. For more illustration of our obtained solutions, some figures for them are given. The power and practical properties of the used method are tested.

scholarly journals Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations

Open Physics ◽  
10.1515/phys-2018-0111 ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 896-909 ◽  
Author(s):  
Dianchen Lu ◽  
Aly R. Seadawy ◽  
Mujahid Iqbal
Keyword(s):  
Solitary Wave ◽  
Nonlinear Partial Differential Equations ◽  
Research Work ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Mapping Method ◽  
New Method ◽  
Auxiliary Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions

AbstractIn this research work, for the first time we introduced and described the new method, which is modified extended auxiliary equation mapping method. We investigated the new exact traveling and families of solitary wave solutions of two well-known nonlinear evaluation equations, which are generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified forms of Camassa-Holm equations. We used a new technique and we successfully obtained the new families of solitary wave solutions. As a result, these new solutions are obtained in the form of elliptic functions, trigonometric functions, kink and antikink solitons, bright and dark solitons, periodic solitary wave and traveling wave solutions. These new solutions show the power and fruitfulness of this new method. We can solve other nonlinear partial differential equations with the use of this method.

scholarly journals The Method of Lines and Adomian Decomposition for Obtaining Solitary Wave Solutions of the KdV Equation

10.5539/apr.v5n3p43 ◽  
2013 ◽  
Vol 5 (3) ◽  
Author(s):  
Mohamed M. Mousa ◽  
Mohamed Reda
Keyword(s):  
Solitary Wave ◽  
Kdv Equation ◽  
Method Of Lines ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
The Method Of Lines ◽  
Adomian Decomposition ◽  
The Kdv Equation

Solitary wave solution of nonlinear Bogoyavlenskii system by variational analysis method

2021 ◽  
Author(s):  
KangLe Wang
Keyword(s):  
Solitary Wave ◽  
Variational Analysis ◽  
Solitary Wave Solution ◽  
Solitary Wave Solutions ◽  
Analysis Method ◽  
Soliton Theory ◽  
Practical Applications ◽  
The Road ◽  
Wave Solutions ◽  
First Time

In this work, the Bogoyavlenskii system (BS) and fractal BS are investigated by variational method for the first time. An efficient and simple scheme is proposed to seek their exact solitary wave solutions, which is called variational analysis method. The novel scheme requires only two steps, making it much attractive in practical applications, and a good result is obtained. This paper cleans up the road to the exact solitions, and it sheds a new light on the soliton theory. Finally, the physical properties of solitary wave solutions obtained are analyzed by some simulation figures.

Lump soliton wave solutions for the (2+1)-dimensional Konopelchenko–Dubrovsky equation and KdV equation

2019 ◽  
Vol 33 (18) ◽  
pp. 1950199 ◽  
Author(s):  
Mostafa M. A. Khater ◽  
Dianchen Lu ◽  
Raghda A. M. Attia
Keyword(s):  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Auxiliary Equation ◽  
Wave Solutions ◽  
All Solutions ◽  
Long Range Interactions ◽  
Weak Scattering ◽  
Novel Method ◽  
Soliton Wave Solutions

This paper studies (2+1)-dimensional Konopelchenko–Dubrovsky equation and (2+1)-dimensional KdV equation via a modified auxiliary equation technique. These two systems describe the connection between the nonlinear weaves with a weak scattering and long-range interactions between the tropical, mid-latitude troposphere, the interaction of equatorial and mid-latitude Rossby waves, respectively. We implement a novel technique to these systems to find analytical traveling wave solutions. The performance of this novel method shows its ability for applying on various nonlinear partial differential equations. All solutions obtained are checked by the Maple software system and verified for its high fidelity.

Solitary wave solutions for the general KDV equation by Adomian decomposition method

2004 ◽  
Vol 154 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Hassan N.A Ismail ◽  
Kamal R Raslan ◽  
Ghada S.E Salem
Keyword(s):  
Solitary Wave ◽  
Decomposition Method ◽  
Kdv Equation ◽  
Adomian Decomposition Method ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Adomian Decomposition

Jacobi Elliptic Function Solutions of Three Coupled Nonlinear Physical Equations

10.1515/zna-2005-0404 ◽  
2005 ◽  
Vol 60 (4) ◽  
pp. 237-244 ◽  
Author(s):  
M. M. Hassan ◽  
A. H. Khater
Keyword(s):  
Solitary Wave ◽  
Elliptic Function ◽  
Nonlinear Partial Differential Equations ◽  
Wave Interaction ◽  
Short Wave ◽  
Jacobi Elliptic Function ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Pacs 02.30 ◽  
Limiting Case

Abstract The Jacobi elliptic function solutions of coupled nonlinear partial differential equations, including the coupled modified KdV (mKdV) equations, long-short-wave interaction system and the Davey- Stewartson (DS) equations, are obtained by using the mixed dn-sn method. The solutions obtained in this paper include the single and the combined Jacobi elliptic function solutions. In the limiting case, the solitary wave solutions of the systems are also given. - PACS: 02.30.Jr; 03.40.Kf; 03.65.Fd

scholarly journals New Exact Solutions for High Dispersive Cubic-Quintic Nonlinear Schrödinger Equation

10.1155/2014/826746 ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yongan Xie ◽  
Shengqiang Tang
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Bifurcation Theory ◽  
Solitary Wave Solutions ◽  
Nonlinear Schrödinger ◽  
Light Pulses ◽  
New Exact Solutions ◽  
Wave Solutions ◽  
First Time ◽  
Quintic Nonlinear Schrödinger Equation

We study a class of high dispersive cubic-quintic nonlinear Schrödinger equations, which describes the propagation of femtosecond light pulses in a medium that exhibits a parabolic nonlinearity law. Applying bifurcation theory of dynamical systems and the Fan sub-equations method, more types of exact solutions, particularly solitary wave solutions, are obtained for the first time.

On solitary wave solutions of a class of nonlinear partial differential equations based on the function1/coshn(αx+βt)

2017 ◽  
Vol 315 ◽  
pp. 372-380 ◽  
Author(s):  
Nikolay K. Vitanov ◽  
Zlatinka I. Dimitrova ◽  
Tsvetelina I. Ivanova
Keyword(s):  
Partial Differential Equations ◽  
Solitary Wave ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Solitary Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions
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