Traveling-wave solutions of the Klein–Gordon equations with M-fractional derivative

Pramana ◽  
2022 ◽  
Vol 96 (1) ◽  
Author(s):  
Alphonse Houwe ◽  
Hadi Rezazadeh ◽  
Ahmet Bekir ◽  
Serge Y Doka
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
Klein Gordon

Explicit, periodic and dispersive soliton solutions to the conformable time-fractional Wu–Zhang system

2021 ◽  
pp. 2150417
Author(s):  
Kalim U. Tariq ◽  
Mostafa M. A. Khater ◽  
Muhammad Younis
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Conformable Fractional Derivative ◽  
Physical Behavior ◽  
Wave Solutions

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified.

scholarly journals Exact Solutions to Time-Fractional Fifth Order KdV Equation by Trial Equation Method Based on Symmetry

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 742 ◽  
Author(s):  
Tao Liu
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Equation Method ◽  
Singular Solutions ◽  
Trial Equation Method ◽  
Wave Solutions ◽  
Fifth Order ◽  
Fkdv Equation

We study a fifth order time-fractional KdV equation (FKdV) under meaning of the conformal fractional derivative. By trial equation method based on symmetry, we construct the abundant exact traveling wave solutions to the FKdV equation. These solutions show rich evolution patterns including solitons, rational singular solutions, periodic and double periodic solutions and so forth. In particular, under the concrete parameters, we give the representations of all these solutions.

Exact Solutions in the Invariant Manifolds of the Generalized Integrable Hénon–Heiles System and Exact Traveling Wave Solutions of Klein–Gordon–Schrödinger Equations

2017 ◽  
Vol 27 (01) ◽  
pp. 1750012 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Periodic Solutions ◽  
Traveling Wave ◽  
Invariant Manifolds ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Wave Solutions ◽  
Klein Gordon

In this paper, we consider the exact explicit solutions for the famous generalized Hénon–Heiles (H–H) system. Corresponding to the three integrable cases, on the basis of the investigation of the dynamical behavior and level curves of the planar dynamical systems, we find all possible explicit exact parametric representations of solutions in the invariant manifolds of equilibrium points in the four-dimensional phase space. These solutions contain quasi-periodic solutions, homoclinic solutions, periodic solutions as well as blow-up solutions. Therefore, we answer the question: what are the flows in the center manifolds and homoclinic manifolds of the generalized Hénon–Heiles (H–H) system. As an application of the above results, we consider the traveling wave solutions for the coupled [Formula: see text]-dimensional Klein–Gordon–Schrödinger Equations with quadratic power nonlinearity.

NEW TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR EVOLUTION EQUATIONS

2011 ◽  
Vol 25 (02) ◽  
pp. 319-327 ◽  
Author(s):  
CHENG-JIE BAI ◽  
HONG ZHAO ◽  
HENG-YING XU ◽  
XIA ZHANG
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mapping Method ◽  
Nonlinear Evolution Equations ◽  
Wave Solutions ◽  
Trigonometric Function Solutions ◽  
Klein Gordon ◽  
Transformation Relation

The deformation mapping method is extended to solve a class of nonlinear evolution equations (NLEEs). Many types of explicit and exact traveling wave solutions, which contain solitary wave solutions, trigonometric function solutions, and Jacobian elliptic function solutions, are obtained by a simple algebraic transformation relation between the solutions of the NLEEs and those of the cubic nonlinear Klein–Gordon (NKG) equation.

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