Solitary wave solutions of selective nonlinear diffusion-reaction equations using homogeneous balance method

Kumar, Rajesh; Kaushal, R. S.; Prasad, Awadhesh

doi:10.1007/s12043-010-0142-4

Cited by 17 publications

(5 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As an application, the traveling wave solutions for Bogoyavlenskii equation and a diffusive predator-prey system which have been constructed using the modified simple equation method. Let us compare between our results obtained in the present article with the well-known results obtained by other authors using different methods as follows: Our results of a diffusive predator-prey system and Bogoyavlenskii equation are new and different from those obtained in [42]- [44] and also our results of the generalized Fisher equation and Burgers-Huxley equation are new and different from those obtained in [45]. It can be concluded that this method is reliable and propose a variety of exact solutions NPDEs.…”

Section: Resultssupporting

confidence: 51%

RETRACTED: <i>The Modified Simple Equation Method and Its Applications in Mathematical Physics and Biology</i>

Abdelrahman¹,

Armanious²,

Zahran³

et al. 2015

AJCM

View full text Add to dashboard Cite

Section: Resultssupporting

confidence: 51%

RETRACTED: <i>The Modified Simple Equation Method and Its Applications in Mathematical Physics and Biology</i>

Abdelrahman¹,

Armanious²,

Zahran³

et al. 2015

AJCM

View full text Add to dashboard Cite

“…In this study, the nonlinear generalized advection-diffusion-reaction (g ADR) equation [5,6] is explored as:…”

Section: Introductionmentioning

confidence: 99%

Symmetry and complexity: a Lie symmetry method to bifurcation, chaos, multistability and soliton solutions of the nonlinear generalized advection-diffusion-reaction equation

Samina,

Jhangeer,

Chen

2024

Phys. Scr.

View full text Add to dashboard Cite

This paper deals into the complexities of nonlinear dynamics within the nonlinear generalized advection-diffusion-reaction equation, which describes intricate transport phenomena involving advection, diffusion,and reaction processes occurring simultaneously. Through the utilization of the Lie symmetry approach,we thoroughly examine this proposed model, transforming the partial differential equation into an ordinarydifferential equation using similarity reduction techniques to facilitate a more comprehensive analysis. Ex-act solutions for this transformed equation are derived employing the extended simplest equation methodand the new extended direct algebraic method. To enhance understanding, contour plots along with 2Dand 3D visualizations of solutions are employed. Additionally, we explore bifurcation and chaotic be-haviours through a qualitative analysis of the model. Phase portraits are meticulously scrutinized acrossvarious parameter values, offering insights into system behavior. Introduction of an external periodicstrength allows us to utilize various tools including time series, 3D, and 2D phase patterns to discernchaotic and quasi-periodic behaviors. Furthermore, a multistability analysis is conducted to examine theimpacts of diverse initial conditions. These findings underscore the efficacy and practicality of the pro-posed methodologies in evaluating soliton solutions and elucidating phase dynamics across a spectrum ofnonlinear models, offering novel perspectives on intricate physical phenomena.

show abstract

“…In current research, the behavior of rogue waves and their solutions for integrable HNLSE are mainly focused. [15][16][17][18][19][20] Several methods for solving the explicit traveling wave solutions of nonlinear evolution equations have been presented, such as the extended modified direct algebraic method, [21] the homotopy perturbation method, [22] the Jocobi elliptic function expansion method, [23] the homogeneous balance method, [24] etc. [25][26][27][28][29][30] In order to obtain solutions of the extended high-order nonlinear Schrodinger equation with variable coefficients, we use a method which is called the (G /G)expansion method [31,32] and has been proposed to construct new bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions to many nonlinear evolution equations.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the rogue waves in the blood based on the high-order NLS equations with variable coefficients*

Yang¹,

Gao²,

Yang³

2021

Chinese Phys. B

View full text Add to dashboard Cite

The research of rogue waves is an advanced field which has important practical and theoretical significances in mathematics, physics, biological fluid mechanics, oceanography, etc. Using the reductive perturbation theory and long wave approximation, the equations governing the movement of blood vessel walls and the flow of blood are transformed into high-order nonlinear Schrödinger (NLS) equations with variable coefficients. The third-order nonlinear Schrödinger equation is degenerated into a completely integrable Sasa–Satsuma equation (SSE) whose solutions can be used to approximately simulate the real rogue waves in the vessels. For the first time, we discuss the conditions for generating rogue waves in the blood vessels and effects of some physiological parameters on the rogue waves. Based on the traveling wave solutions of the fourth-order nonlinear Schrödinger equation, we analyze the effects of the higher order terms and the initial deformations of the blood vessel on the wave propagation and the displacement of the tube wall. Our results reveal that the amplitude of the rogue waves are proportional to the initial stretching ratio of the tube. The high-order nonlinear and dispersion terms lead to the distortion of the wave, while the initial deformation of the tube wall will influence the wave amplitude and wave steepness.

show abstract

Solitary wave solutions of selective nonlinear diffusion-reaction equations using homogeneous balance method

Cited by 17 publications

References 34 publications

RETRACTED: <i>The Modified Simple Equation Method and Its Applications in Mathematical Physics and Biology</i>

RETRACTED: <i>The Modified Simple Equation Method and Its Applications in Mathematical Physics and Biology</i>

Symmetry and complexity: a Lie symmetry method to bifurcation, chaos, multistability and soliton solutions of the nonlinear generalized advection-diffusion-reaction equation

Analysis of the rogue waves in the blood based on the high-order NLS equations with variable coefficients*

Contact Info

Product

Resources

About

Solitary wave solutions of selective nonlinear diffusion-reaction equations using homogeneous balance method

Cited by 17 publications

References 34 publications

RETRACTED: &lt;i&gt;The Modified Simple Equation Method and Its Applications in Mathematical Physics and Biology&lt;/i&gt;

RETRACTED: &lt;i&gt;The Modified Simple Equation Method and Its Applications in Mathematical Physics and Biology&lt;/i&gt;

Symmetry and complexity: a Lie symmetry method to bifurcation, chaos, multistability and soliton solutions of the nonlinear generalized advection-diffusion-reaction equation

Analysis of the rogue waves in the blood based on the high-order NLS equations with variable coefficients*

Contact Info

Product

Resources

About

RETRACTED: <i>The Modified Simple Equation Method and Its Applications in Mathematical Physics and Biology</i>

RETRACTED: <i>The Modified Simple Equation Method and Its Applications in Mathematical Physics and Biology</i>