scholarly journals New Travelling Wave Solution-Based New Riccati Equation for Solving KdV and Modified KdV Equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Hadi Rezazadeh ◽  
Alper Korkmaz ◽  
Abdelfattah EL Achab ◽  
Waleed Adel ◽  
Ahmet Bekir
Keyword(s):  
Large Family ◽  
Algebraic Method ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Hyperbolic Functions ◽  
Kdv Equations ◽  
De Vries ◽  
Wave Forms ◽  
Modified Kdv Equations ◽  
Different Characteristics

AbstractA large family of explicit exact solutions to both Korteweg- de Vries and modified Korteweg- de Vries equations are determined by the implementation of the new extended direct algebraic method. The procedure starts by reducing both equations to related ODEs by compatible travelling wave transforms. The balance between the highest degree nonlinear and highest order derivative terms gives the degree of the finite series. Substitution of the assumed solution and some algebra results in a system of equations are found. The relation between the parameters is determined by solving this system. The solutions of travelling wave forms determined by the application of the approach are represented in explicit functions of some generalized trigonometric and hyperbolic functions and exponential function. Some more solutions with different characteristics are also found.

scholarly journals An Expansion Based on Sine-Gordon Equation to Solve KdV and modified KdV Equations in Conformable Fractional Forms

2017 ◽  
Author(s):  
Ozlem Ersoy Hepson ◽  
Alper Korkmaz ◽  
Kamyar Hosseini ◽  
Hadi Rezazadeh ◽  
Mostafa Eslami
Keyword(s):  
Exact Solutions ◽  
Gordon Equation ◽  
Expansion Method ◽  
Sine Gordon Equation ◽  
Hyperbolic Functions ◽  
Kdv Equations ◽  
De Vries ◽  
Balance Technique ◽  
Modified Kdv Equations ◽  
Complex Valued

An expansion method based on time fractional Sine-Gordon equation is implemented to construct some real and complex valued exact solutions to the Korteweg-de Vries and modified Korteweg-de Vries equation in time fractional forms. Compatible fractional traveling wave transform plays a key role to be able to apply homogeneous balance technique to set the predicted solution. The relation between trigonometric and hyperbolic functions based on fractional Sine-Gordon equation allows to form the exact solutions with multiplication of powers of hyperbolic functions.

scholarly journals Application of mathematical methods for the non-linear seventh order Sawada-Kotera-Ito dynamical wave equation

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 2081-2093 ◽  
Author(s):  
Wafaa Aibarakati ◽  
Aly Seadaw ◽  
Noufe Aljahdaly
Keyword(s):  
Wave Equation ◽  
Algebraic Method ◽  
Travelling Wave ◽  
Physical Structure ◽  
Travelling Wave Solutions ◽  
Mathematical Methods ◽  
Kdv Equations ◽  
Article Deal ◽  
Seventh Order ◽  
Stable Solutions

This article deal with finding travelling wave solutions for the 7th order Sawada-Kotera-Ito dynamical wave equation which describes the evolution of steeper waves of shorter wavelength than KdV equations using modified extended direct algebraic method. The new solutions derived have various physical structure, we also give graphic representation of the exact and stable solutions.

scholarly journals Conservation law and exact solutions for ion acoustic waves in a magnetized quantum plasma

2016 ◽  
Vol 5 (3) ◽  
pp. 138 ◽  
Author(s):  
Omar El-Kalaawy ◽  
Rafat Ibrahim ◽  
Lamyaa Sadek
Keyword(s):  
Acoustic Waves ◽  
Traveling Wave Solutions ◽  
Algebraic Method ◽  
Magnetized Plasma ◽  
Quantum Plasma ◽  
Kdv Equations ◽  
Ion Acoustic ◽  
De Vries ◽  
Analytical Phase ◽  
Two Sides

The head on collision of ion- acoustic solitary waves (IASWs) in a magnetized plasma are considered. The two- sides Korteweg-de Vries (KdV) equations in generic case as well as the two- sides modified Korteweg-de Vries (mKdV) equations in a special case are obtained.The analytical phase shifts and the trajectories after the Head-on collisions of two IASWs in a three species quantum plasma are derived by using the extended version of Poincare-Lighthill-Kuo (PLK) method for both the situations. The conservation laws for KdV and mKdV equations are obtained. By applying the extended direct algebraic method, we found the traveling wave solutions for the two-sides KdV and mKdV equations.

An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

2011 ◽  
Vol 141 (6) ◽  
pp. 1141-1173 ◽  
Author(s):  
Jared C. Bronski ◽  
Mathew A. Johnson ◽  
Todd Kapitula
Keyword(s):  
Blow Up ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Classical Mechanics ◽  
Index Theorem ◽  
Travelling Wave ◽  
Kdv Equations ◽  
The Kdv Equation ◽  
De Vries ◽  
The Stability

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

scholarly journals Mapping Methods to Solve a Modified Korteweg-de Vries Type Equation

2015 ◽  
Vol 20 (2) ◽  
pp. 42 ◽  
Author(s):  
Edamana. V. Krishnan
Keyword(s):  
Vries Equation ◽  
Type Equation ◽  
Elliptic Functions ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Hyperbolic Functions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Singular Wave

In this paper, we employ mapping methods to construct exact travelling wave solutions for a modified Korteweg-de Vries equation. We have derived periodic wave solutions in terms of Jacobi elliptic functions, kink solutions and singular wave solutions in terms of hyperbolic functions.  

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