Kinetics of phase separation in Fe–Cr–X (X =Mo, Cu) ternary alloys — a dynamical wave study

2021 ◽  
pp. 2150220
Author(s):  
M. Younis ◽  
Aly R. Seadawy ◽  
M. Bilal ◽  
S. U. Rehman ◽  
S. Latif ◽  
...  
Keyword(s):  
Phase Separation ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Ternary Alloys ◽  
Unknown Parameters ◽  
Wave Solutions ◽  
Trigonometric Function Solutions ◽  
Kinetics Of

Ternary alloys of Fe are very important materials having good corrosion resistance and are also famous for several high-temperature applications. The dynamical behavior of exact traveling waves for the kinetics of phase separation in Fe–Cr–X (X[Formula: see text]Mo, Cu) ternary alloys is modelled by convective-diffusive Cahn–Hilliard (CH) equation. A variety of nonlinear dynamical exact and solitary wave structures are extracted in several forms like rational, hyperbolic, trigonometric function solutions by the utilization of a sound computational integration tool, i.e., [Formula: see text]-model expansion method. Besides, we also secure mixed combined solitons and singular periodic wave solutions with unknown parameters. Moreover, the constraint conditions observed during derivation lead to substantial solutions. The findings elucidate that the governing model theoretically possesses significantly rich structures of exact traveling wave solutions. Hence, our technique via fortification of symbolic computations provides an active and potent mathematical implementation for solving diverse benevolent nonlinear wave problems.

scholarly journals Exact Solutions to the Sharma-Tasso-Olver Equation by Using ImprovedG′/G-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yinghui He ◽  
Shaolin Li ◽  
Yao Long
Keyword(s):  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Periodic Traveling Waves ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

This paper is concerned with a double nonlinear dispersive equation: the Sharma-Tasso-Olver equation. We propose an improvedG′/G-expansion method which is employed to investigate the solitary and periodic traveling waves of this equation. As a result, some new traveling wave solutions involving hyperbolic functions, the trigonometric functions, are obtained. When the parameters are taken as special values, the solitary wave solutions are derived from the hyperbolic function solutions, and the periodic wave solutions are derived from the trigonometric function solutions. The improvedG′/G-expansion method is straightforward, concise and effective and can be applied to other nonlinear evolution equations in mathematical physics.

ON A CLASS OF SINGULAR NONLINEAR TRAVELING WAVE EQUATIONS (II): AN EXAMPLE OF GCKdV EQUATIONS

2009 ◽  
Vol 19 (06) ◽  
pp. 1995-2007 ◽  
Author(s):  
JIBIN LI ◽  
YI ZHANG ◽  
XIAOHUA ZHAO
Keyword(s):  
Traveling Wave ◽  
Wave Equations ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Kink Wave ◽  
Coupled Kdv Equations

By using the method of dynamical systems, we continuously study the dynamical behavior for the first class of singular nonlinear traveling wave systems. As an example, the traveling wave solutions for a generalized coupled KdV equations are discussed. Exact explicit parametric representations of solitary wave solutions, periodic wave solutions and kink wave solutions are given.

Bifurcations and Exact Traveling Wave Solutions of a Modified Nonlinear Schrödinger Equation

2016 ◽  
Vol 26 (06) ◽  
pp. 1650106 ◽  
Author(s):  
KitIan Kou ◽  
Jibin Li
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
One Dimensional ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Planar Dynamical Systems

In this paper, we consider two singular nonlinear planar dynamical systems created from the studies of one-dimensional bright and dark spatial solitons for one-dimensional beams in a nonlocal Kerr-like media. On the basis of the investigation of the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, peakon and periodic peakons, compacton solutions, etc.) under different parameter conditions.

Bifurcations and Exact Traveling Wave Solutions of Degenerate Coupled Multi-KdV Equations

2016 ◽  
Vol 26 (03) ◽  
pp. 1650045 ◽  
Author(s):  
Jibin Li ◽  
Fengjuan Chen
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Kink Wave ◽  
Multiplicity Formula ◽  
Planar Dynamical Systems

In this paper, we consider the degenerate coupled multi-KdV equations. Depending on the coupled multiplicity [Formula: see text], the study of the traveling wave solutions for this model derives a series of planar dynamical systems. We consider the cases of [Formula: see text] On the basis of the investigation on the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions) under different parameter conditions.

A generalization of (G′/G)-expansion method and its application to nonlinear reaction–diffusion equations arising in mathematical biology

2014 ◽  
Vol 07 (03) ◽  
pp. 1450025 ◽  
Author(s):  
A. Jabbari ◽  
J. Manafian Heris ◽  
H. Kheiri ◽  
A. Bekir
Keyword(s):  
Mathematical Biology ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Expansion Method ◽  
Periodic Wave ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Hyperbolic Function ◽  
Wave Solutions ◽  
Nonlinear Reaction

In this paper, by introducing a proper transformation, the (G′/G)-expansion method is further extended into the nonlinear reaction–diffusion equations in mathematical biology whose balancing numbers may be negative integer. As a result, hyperbolic function solutions and trigonometric function solutions with free parameters are obtained. When the parameters are taken as special values the solitary wave solutions and the periodic wave solutions are also derived from the traveling wave solutions. Moreover, it is observed that the suggested techniques is compatible of such problems.

On the Exact Traveling Wave Solutions of (2 + 1)-Dimensional Higher Order Broer–Kaup Equation

2014 ◽  
Vol 24 (01) ◽  
pp. 1450007 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Higher Order ◽  
Solution Component ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave ◽  
Wave Profiles

In this paper, we study the dynamical behavior and exact parametric representations of all traveling wave solutions for (2 + 1)-dimensional higher order Broer–Kaup equation. By using the method of dynamical systems, under different parametric conditions, for the solution component U, exact monotonic and nonmonotonic kink wave solutions, two-peak wave solutions, periodic wave solutions, as well as unbounded traveling wave solutions are obtained. Exact wave profiles of traveling wave solutions for all solution components U, V, W, P are shown.

Breather-wave, periodic-wave and traveling-wave solutions for a (2 + 1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation for an incompressible fluid

2021 ◽  
pp. 2150261
Author(s):  
Yuan Shen ◽  
Bo Tian ◽  
Chen-Rong Zhang ◽  
He-Yuan Tian ◽  
Shao-Hua Liu
Keyword(s):  
Incompressible Fluid ◽  
Theta Function ◽  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Polynomial Expansion Method

In this paper, the investigation is conducted on a (2 + 1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation for an incompressible fluid. Via the Riemann theta function, periodic-wave solutions are derived, and breather-wave solutions are constructed with the aid of the extended homoclinic test approach. Based on the polynomial expansion method, several traveling-wave solutions are derived. Besides, we observe that the amplitude of the breather keeps unchanged during the propagation and the traveling wave which is kink shaped propagates stably. Furthermore, we analyze the transition between the periodic-wave and soliton solutions, which implies that the periodic-wave solutions tend to the soliton solutions via a limiting procedure.

Exact Traveling Wave Solutions of the Krichever–Novikov Equation: A Dynamical System Approach

2017 ◽  
Vol 27 (04) ◽  
pp. 1750058 ◽  
Author(s):  
KitIan Kou ◽  
Jibin Li
Keyword(s):  
Traveling Wave ◽  
Invariant Manifolds ◽  
Solitary Wave Solution ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Painlevé Property ◽  
Novikov Equation

In this paper, we show that to find the traveling wave solutions for the Krichever–Novikov equation, we only need to consider a spatial form F-VI of the fourth-order differential equations in the polynomial class having the Painlevé property given by [Cosgrove, 2000]. By using the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions in some two-dimensional invariant manifolds, various exact solutions such as solitary wave solution, periodic wave solutions, quasi-periodic wave solutions and uncountably infinitely many unbounded wave solutions are obtained.

scholarly journals New exact traveling wave solutions to the (2+1)-dimensional Chiral nonlinear Schrödinger equation

2021 ◽  
Author(s):  
Hadi Rezazadeh ◽  
Muhammad Younis ◽  
Shafqat Ur-Rehman ◽  
Muhammad Bilal ◽  
Usman Younas ◽  
...  
Keyword(s):  
Applied Sciences ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Research Work ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Unknown Parameters ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Significant Addition

In this research work, we successfully construct various kinds of exact traveling wave solutions such as trigonometric like, singular and periodic wave solutions as well as hyperbolic solutions to the (2+1)-dimensional Chiral nonlinear Schröginger equation (CNLSE) which is used as a governing equation to discuss the wave in the quantum field theory. The mechanisms which are used to obtain these solutions are extended rational sine- cosine/sinh-cosh and the constraint conditions for the existence of valid solutions are also given. The attained results exhibit that the proposed techniques are the significant addition for exploring several types of nonlinear partial differential equations in applied sciences. Moreover, 3D and contour profiles are depicted for showing the physical behaviour of the reported solutions by setting suitable values of unknown parameters

Bifurcations of Traveling Wave Solutions of a Nonlinear Wave Model Created by Diffraction in Periodic Media

2016 ◽  
Vol 26 (02) ◽  
pp. 1650032
Author(s):  
Jibin Li ◽  
Fengjuan Chen
Keyword(s):  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Wave Model ◽  
Periodic Wave ◽  
Periodic Media ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Straight Line ◽  
Planar Dynamical Systems

In this paper, we consider a model created by diffraction in periodic media. The study of the traveling wave solutions for this model derives a planar dynamical system with a singular straight line. On the basis of the investigation of the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, periodic peakon solutions, compactons, etc.) under different parameter conditions.

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