An Extension of the Stability Index for Traveling-Wave Solutions and Its Application to Bifurcations

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.

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Comparison exact and numerical simulation of the traveling wave solution in nonlinear dynamics

International Journal of Modern Physics B ◽

10.1142/s0217979220502823 ◽

2020 ◽

Vol 34 (29) ◽

pp. 2050282

Author(s):

Asıf Yokuş ◽

Doğan Kaya

Keyword(s):

Traveling Wave ◽

Numerical Solutions ◽

Traveling Wave Solutions ◽

Transformation Method ◽

Mkdv Equation ◽

Traveling Wave Solution ◽

Von Neumann ◽

Wave Solutions ◽

De Vries ◽

The Stability

The traveling wave solutions of the combined Korteweg de Vries-modified Korteweg de Vries (cKdV-mKdV) equation and a complexly coupled KdV (CcKdV) equation are obtained by using the auto-Bäcklund Transformation Method (aBTM). To numerically approximate the exact solutions, the Finite Difference Method (FDM) is used. In addition, these exact traveling wave solutions and numerical solutions are compared by illustrating the tables and figures. Via the Fourier–von Neumann stability analysis, the stability of the FDM with the cKdV–mKdV equation is analyzed. The [Formula: see text] and [Formula: see text] norm errors are given for the numerical solutions. The 2D and 3D figures of the obtained solutions to these equations are plotted.

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New exact traveling wave solutions of biological population model via the extended rational sinh-cosh method and the modified Khater method

Modern Physics Letters B ◽

10.1142/s021798491950338x ◽

2019 ◽

Vol 33 (28) ◽

pp. 1950338 ◽

Cited By ~ 30

Author(s):

Hadi Rezazadeh ◽

Alper Korkmaz ◽

Mostafa M. A. Khater ◽

Mostafa Eslami ◽

Dianchen Lu ◽

...

Keyword(s):

Exact Solutions ◽

Traveling Wave ◽

Population Model ◽

Stability Property ◽

Traveling Wave Solutions ◽

Algebraic Methods ◽

New Exact Solutions ◽

Wave Solutions ◽

Biological Population ◽

The Stability

In this paper, the extended rational sinh-cosh method (ERSCM) and modified Khater method are applied to the biological population model to derive new exact solutions. Moreover, the stability property of some obtained solutions is discussed to show the ability of them for using in the model’s applications. Implementation of the direct algebraic methods, the equations derived by substitution of the predicted solution are solved. It is significant to point out that new traveling wave solutions are found. The present methods are easy to employ and sufficient to determine the solutions.

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The stability of traveling wave solutions for a diffusive competition system of three species

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2017.10.013 ◽

2018 ◽

Vol 459 (1) ◽

pp. 564-576 ◽

Cited By ~ 2

Author(s):

Chueh-Hsin Chang

Keyword(s):

Traveling Wave ◽

Traveling Wave Solutions ◽

Competition System ◽

Wave Solutions ◽

The Stability

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Stability Analysis and Assortment of Exact Traveling Wave Solutions for the (2+1)-Dimensional Boiti-Leon-Pempinelli System

Iraqi Journal of Science ◽

10.24996/ijs.2021.62.7.29 ◽

2021 ◽

pp. 2393-2400

Author(s):

Mizal H. Alobaidi ◽

Wafaa M. Taha ◽

Ali H. Hazza ◽

Pelumi E. Oguntunde

Keyword(s):

Stability Analysis ◽

Traveling Wave ◽

Physical Meaning ◽

Function Method ◽

Traveling Wave Solutions ◽

Exact Results ◽

Exact Traveling Wave Solutions ◽

Wave Solutions ◽

The Stability

In this research, the Boiti–Leon–Pempinelli (BLP) system was used to understand the physical meaning of exact and solitary traveling wave solutions. To establish modern exact results, considered. In addition, the results obtained were compared with those obtained by using other existing methods, such as the standard hyperbolic tanh function method, and the stability analysis for the results was discussed.

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Traveling Wave Solutions of the Benjamin-Bona-Mahony Water Wave Equations

Abstract and Applied Analysis ◽

10.1155/2014/926838 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 10

Author(s):

A. R. Seadawy ◽

A. Sayed

Keyword(s):

Traveling Wave ◽

Water Waves ◽

Water Wave ◽

Wave Equations ◽

Traveling Wave Solutions ◽

Dispersive Media ◽

Wave Solutions ◽

De Vries ◽

The Stability

The modeling of unidirectional propagation of long water waves in dispersive media is presented. The Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations are derived from water waves models. New traveling solutions of the KdV and BBM equations are obtained by implementing the extended direct algebraic and extended sech-tanh methods. The stability of the obtained traveling solutions is analyzed and discussed.

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Stability of Traveling Waves to the Lotka-Volterra Competition Model

Complexity ◽

10.1155/2019/6569520 ◽

2019 ◽

Vol 2019 ◽

pp. 1-11

Author(s):

Ahmad Alhasanat ◽

Chunhua Ou

Keyword(s):

Global Stability ◽

Comparison Principle ◽

Traveling Waves ◽

Traveling Wave ◽

Traveling Wave Solutions ◽

Functional Space ◽

Cooperative System ◽

Wave Solutions ◽

Diffusive Model ◽

The Stability

In this paper, the stability of traveling wave solutions to the Lotka-Volterra diffusive model is investigated. First, we convert the model into a cooperative system by a special transformation. The local and the global stability of the traveling wavefronts are studied in a weighted functional space. For the global stability, comparison principle together with the squeezing technique is applied to derive the main results.

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Traveling Wave Solutions of a Delayed Cooperative System

Mathematics ◽

10.3390/math7030269 ◽

2019 ◽

Vol 7 (3) ◽

pp. 269 ◽

Cited By ~ 6

Author(s):

Xue-Shi Li ◽

Shuxia Pan

Keyword(s):

Asymptotic Behavior ◽

Traveling Wave ◽

Traveling Wave Solutions ◽

Sufficient Condition ◽

Cooperative System ◽

Wave Solutions ◽

Existence And Nonexistence ◽

Functional Differential ◽

The Stability ◽

Functional Differential System

This paper deals with the dynamics of a delayed cooperative system without quasimonotonicity. Using the contracting rectangles, we obtain a sufficient condition on the stability of the unique positive steady state of the functional differential system. When the spatial domain is whole R , the existence and nonexistence of traveling wave solutions are investigated, during which the asymptotic behavior is investigated by the contracting rectangles.

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