The Multiple Exp-Function Method and the Linear Superposition Principle for Solving the (2+1)-Dimensional Calogero–Bogoyavlenskii–Schiff Equation

2015 ◽  
Vol 70 (9) ◽  
pp. 775-779 ◽  
Author(s):  
Elsayed M.E. Zayed ◽  
Abdul-Ghani Al-Nowehy
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Function Method ◽  
Superposition Principle ◽  
Solitary Wave Solutions ◽  
Linear Superposition ◽  
Wave Solutions ◽  
Linear Superposition Principle

AbstractIn this article, the multiple exp-function method and the linear superposition principle are employed for constructing the exact solutions and the solitary wave solutions for the (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. With help of Maple and by using the multiple exp-method, we can get exact explicit one-wave, two-wave, and three-wave solutions, which include one-soliton-, two-soliton-, and three-soliton-type solutions. Furthermore, we apply the linear superposition principle to find n-wave solutions of the CBS equation. Two cases with specific values of the involved parameters are plotted for each two-wave and three-wave solutions.

Solitary wave solutions of (2+1)-dimensional Maccari system

2021 ◽  
pp. 2150391
Author(s):  
Ghazala Akram ◽  
Naila Sajid
Keyword(s):  
Nonlinear Optics ◽  
Solitary Wave ◽  
Plasma Physics ◽  
Exact Solutions ◽  
Function Method ◽  
Expansion Method ◽  
Periodic Wave ◽  
Hyperbolic Function ◽  
Solitary Wave Solutions ◽  
Wave Solutions

In this article, three mathematical techniques have been operationalized to discover novel solitary wave solutions of (2+1)-dimensional Maccari system, which also known as soliton equation. This model equation is usually of applicative relevance in hydrodynamics, nonlinear optics and plasma physics. The [Formula: see text] function, the hyperbolic function and the [Formula: see text]-expansion techniques are used to obtain the novel exact solutions of the (2+1)-dimensional Maccari system (arising in nonlinear optics and in plasma physics). Many novel solutions such as periodic wave solutions by [Formula: see text] function method, singular, combined-singular and periodic solutions by hyperbolic function method, hyperbolic, rational and trigonometric solutions by [Formula: see text]-expansion method are obtained. The exact solutions are shown through 3D graphics which present the movement of the obtained solutions.

scholarly journals New Exact Solutions to the KdV-Burgers-Kuramoto Equation with the Exp-Function Method

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Jae-Myoung Kim ◽  
Changbum Chun
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Function Method ◽  
Expansion Method ◽  
Solitary Wave Solutions ◽  
New Exact Solutions ◽  
Wave Solutions ◽  
Truncated Painlevé Expansion

Based on the characteristics of the truncated Painlevé expansion method and the Exp-function method, new generalized solitary wave solutions are constructed for the KdV-Burgers-Kuramoto equation, which cannot be directly constructed from the Exp-function method. This work highlights the power of the Exp-function method in providing generalized solitary wave solutions of different physical structures.

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

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.

scholarly journals New Solitary and Periodic Wave Solutions of (n + 1)-Dimensional Fractional Order Equations Modeling Fluid Dynamics

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2017
Author(s):  
Sadullah Bulut ◽  
Mesut Karabacak ◽  
Hijaz Ahmad ◽  
Sameh Askar
Keyword(s):  
Solitary Wave ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Wave Model ◽  
Expansion Method ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
New Exact Solutions ◽  
Wave Solutions ◽  
Symmetry Properties

In this study, first, fractional derivative definitions in the literature are examined and their disadvantages are explained in detail. Then, it seems appropriate to apply the (G′G)-expansion method under Atangana’s definition of β-conformable fractional derivative to obtain the exact solutions of the space–time fractional differential equations, which have attracted the attention of many researchers recently. The method is applied to different versions of (n+1)-dimensional Kadomtsev–Petviashvili equations and new exact solutions of these equations depending on the β parameter are acquired. If the parameter values in the new solutions obtained are selected appropriately, 2D and 3D graphs are plotted. Thus, the decay and symmetry properties of solitary wave solutions in a nonlocal shallow water wave model are investigated. It is also shown that all such solitary wave solutions are symmetrical on both sides of the apex. In addition, a close relationship is established between symmetric and propagated wave solutions.

scholarly journals Lie Symmetry Analysis and Dynamics of Exact Solutions of the (2+1)-Dimensional Nonlinear Sharma–Tasso–Olver Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Sachin Kumar ◽  
Ilyas Khan ◽  
Setu Rani ◽  
Behzad Ghanbari
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Main Concern ◽  
Solitary Wave Solutions ◽  
Lie Symmetry Analysis ◽  
Soliton Theory ◽  
Symmetry Reductions ◽  
Wave Solutions

In soliton theory, the dynamics of solitary wave solutions may play a crucial role in the fields of mathematical physics, plasma physics, biology, fluid dynamics, nonlinear optics, condensed matter physics, and many others. The main concern of this present article is to obtain symmetry reductions and some new explicit exact solutions of the (2 + 1)-dimensional Sharma–Tasso–Olver (STO) equation by using the Lie symmetry analysis method. The infinitesimals for the STO equation were achieved under the invariance criteria of Lie groups. Then, the two stages of symmetry reductions of the governing equation are obtained with the help of an optimal system. Meanwhile, this Lie symmetry method will reduce the STO equation into new partial differential equations (PDEs) which contain a lesser number of independent variables. Based on numerical simulation, the dynamical characteristics of the solitary wave solutions illustrate multiple-front wave profiles, solitary wave solutions, kink wave solitons, oscillating periodic solitons, and annihilation of parabolic wave structures via 3D plots.

scholarly journals New exact solutions of the(2+1)-dimensional Broer-Kaup equation by the consistent Riccati expansion method

2017 ◽  
Vol 21 (4) ◽  
pp. 1783-1788 ◽  
Author(s):  
Ying Jiang ◽  
Da-Quan Xian ◽  
Zheng-De Dai
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Expansion Method ◽  
High Dimensional ◽  
Solitary Wave Solutions ◽  
New Exact Solutions ◽  
Dynamical Behaviors ◽  
Wave Solutions ◽  
Linear Waves ◽  
Consistent Riccati Expansion

In this work, we study the (2+1)-D Broer-Kaup equation. The composite periodic breather wave, the exact composite kink breather wave and the solitary wave solutions are obtained by using the coupled degradation technique and the consistent Riccati expansion method. These results may help us to investigate some complex dynamical behaviors and the interaction between composite non-linear waves in high dimensional models

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