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Article
Publication date: 17 June 2020

Gangwei Wang and Abdul-Majid Wazwaz

The purpose of this paper is to concern with introducing symmetry analysis to the extended Sakovich equation.

Abstract

Purpose

The purpose of this paper is to concern with introducing symmetry analysis to the extended Sakovich equation.

Design/methodology/approach

The newly developed Sakovich equation has been handled by using the Lie symmetries via using the Lie group method.

Findings

The developed extended Sakovich model exhibit symmetries and invariant solutions.

Research limitations/implications

The present study is to address the two main motivations: the study of symmetry analysis and the study of soliton solutions of the extended Sakovich equation.

Practical implications

The work introduces symmetry analysis to the Painlevé-integrable extended Sakovich equation.

Social implications

The work presents useful symmetry algorithms for handling new integrable equations.

Originality/value

The paper presents an original work with symmetry analysis and shows useful findings.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 25 June 2019

Abdul-Majid Wazwaz and Gui-Qiong Xu

The purpose of this paper is to develop a new time-dependent KdV6 equation. The authors derive multiple soliton solutions and multiple complex soliton solutions for a…

Abstract

Purpose

The purpose of this paper is to develop a new time-dependent KdV6 equation. The authors derive multiple soliton solutions and multiple complex soliton solutions for a time-dependent equation.

Design/methodology/approach

The newly developed time-dependent model has been handled by using the Hirota’s direct method. The authors also use the complex Hirota’s criteria for deriving multiple complex soliton solutions.

Findings

The examined extension of the KdV6 model exhibits complete integrability for any analytic time-dependent coefficient.

Research limitations/implications

The paper presents a new efficient algorithm for constructing extended models which give a variety of multiple real and complex soliton solutions.

Practical implications

The paper introduced a new time-dependent KdV6 equation, where integrability is emphasized for any analytic time-dependent function.

Social implications

The findings are new and promising. Multiple real and multiple complex soliton solutions were formally derived.

Originality/value

This is an entirely new work where a new time-dependent KdV6 equation is established. This is the first time that the KdV6 equation is examined as a time-dependent equation. Moreover, the complete integrability of this newly developed equation is emphasized via using Painlevé test.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 11
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 7 May 2021

Sachin Kumar, Rajesh Kumar Gupta and Pinki Kumari

This study aims to find the symmetries and conservation laws of a new Painlevé integrable Broer-Kaup (BK) system with variable coefficients. This system is an extension of…

Abstract

Purpose

This study aims to find the symmetries and conservation laws of a new Painlevé integrable Broer-Kaup (BK) system with variable coefficients. This system is an extension of dispersive long wave equations. As the system is generalized and new, it is essential to explore some of its possible aspects such as conservation laws, symmetries, Painleve integrability, etc.

Design/methodology/approach

This paper opted for an exploratory study of a new Painleve integrable BK system with variable coefficients. Some analytic solutions are obtained by Lie classical method. Then the conservation laws are derived by multiplier method.

Findings

This paper presents a complete set of point symmetries without any restrictions on choices of coefficients, which subsequently yield analytic solutions of the series and solitary waves. Next, the authors derive every admitted non-trivial conservation law that emerges from multipliers.

Research limitations/implications

The authors have found that the considered system is likely to be integrable. So some other aspects such as Lax pair integrability, solitonic behavior and Backlund transformation can be analyzed to check the complete integrability further.

Practical implications

The authors develop a time-dependent Painleve integrable long water wave system. The model represents more specific data than the constant system. The authors presented analytic solutions and conservation laws.

Originality/value

The new time-dependent Painleve integrable long water wave system features some interesting results on symmetries and conservation laws.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 12
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 27 September 2019

Abdul-Majid Wazwaz

The purpose of this paper is to introduce two new Painlevé-integrable extended Sakovich equations with (2 + 1) and (3 + 1) dimensions. The author obtains multiple soliton…

Abstract

Purpose

The purpose of this paper is to introduce two new Painlevé-integrable extended Sakovich equations with (2 + 1) and (3 + 1) dimensions. The author obtains multiple soliton solutions and multiple complex soliton solutions for these three models.

Design/methodology/approach

The newly developed Sakovich equations have been handled by using the Hirota’s direct method. The author also uses the complex Hirota’s criteria for deriving multiple complex soliton solutions.

Findings

The developed extended Sakovich models exhibit complete integrability in analogy with the original Sakovich equation.

Research limitations/implications

This paper is to address these two main motivations: the study of the integrability features and solitons solutions for the developed methods.

Practical implications

This paper introduces two Painlevé-integrable extended Sakovich equations which give real and complex soliton solutions.

Social implications

This paper presents useful algorithms for constructing new integrable equations and for handling these equations.

Originality/value

This paper gives two Painlevé-integrable extended equations which belong to second-order PDEs. The two developed models do not contain the dispersion term uxxx. This paper presents an original work with newly developed integrable equations and shows useful findings.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 12 May 2021

Abdul-Majid Wazwaz

This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author…

Abstract

Purpose

This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space.

Design/methodology/approach

The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense.

Findings

The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium.

Research limitations/implications

The author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method.

Practical implications

The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition.

Social implications

The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions.

Originality/value

The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 4 September 2019

Abdul-Majid Wazwaz

The purpose of this paper is concerned with investigating three integrable shallow water waves equations with time-dependent coefficients. The author obtains multiple…

Abstract

Purpose

The purpose of this paper is concerned with investigating three integrable shallow water waves equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for these three models.

Design/methodology/approach

The newly developed equations with time-dependent coefficients have been handled by using Hirota’s direct method. The author also uses the complex Hirota’s criteria for deriving multiple complex soliton solutions.

Findings

The developed integrable models exhibit complete integrability for any analytic time-dependent coefficients defined though compatibility conditions.

Research limitations/implications

The paper presents an efficient algorithm for handling time-dependent integrable equations with analytic time-dependent coefficients.

Practical implications

This study introduces three new integrable shallow water waves equations with time-dependent coefficients. These models represent more specific data than the related equations with constant coefficients. The author shows that integrable equations with time-dependent coefficients give real and complex soliton solutions.

Social implications

The paper presents useful algorithms for finding integrable equations with time-dependent coefficients.

Originality/value

The paper presents an original work with a variety of useful findings.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 30 October 2020

Abdul-Majid Wazwaz

This study aims to develop a new (3 + 1)-dimensional Painlevé-integrable extended Vakhnenko–Parkes equation. The author formally derives multiple soliton solutions for…

Abstract

Purpose

This study aims to develop a new (3 + 1)-dimensional Painlevé-integrable extended Vakhnenko–Parkes equation. The author formally derives multiple soliton solutions for this developed model.

Design/methodology/approach

The study used the simplified Hirota’s method for deriving multiple soliton solutions.

Findings

The study finds that the developed (3 + 1)-dimensional Vakhnenko–Parkes model exhibits complete integrability in analogy with the standard Vakhnenko–Parkes equation.

Research limitations/implications

This study addresses the integrability features of this model via using the Painlevé analysis. The study also reports multiple soliton solutions for this equation by using the simplified Hirota’s method.

Practical implications

The work reports extension of the (1 + 1)-dimensional standard equation to a (3 + 1)-dimensional model.

Social implications

The work presents useful algorithms for constructing new integrable equations and for handling these equations.

Originality/value

The paper presents an original work with newly developed integrable equation and shows useful findings.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 11 January 2021

Abdul-Majid Wazwaz

This paper aims to develop a new (3 + 1)-dimensional Painlevée-integrable extended Sakovich equation. This paper formally derives multiple soliton solutions for this…

Abstract

Purpose

This paper aims to develop a new (3 + 1)-dimensional Painlevée-integrable extended Sakovich equation. This paper formally derives multiple soliton solutions for this developed model.

Design/methodology/approach

This paper uses the simplified Hirota’s method for deriving multiple soliton solutions.

Findings

This paper finds that the developed (3 + 1)-dimensional Sakovich model exhibits complete integrability in analogy with the standard Sakovich equation.

Research limitations/implications

This paper addresses the integrability features of this model via using the Painlevée analysis. This paper reports multiple soliton solutions for this equation by using the simplified Hirota’s method.

Practical implications

The study reports three non-linear terms added to the standard Sakovich equation.

Social implications

The study presents useful algorithms for constructing new integrable equations and for handling these equations.

Originality/value

The paper reports a new Painlevée-integrable extended Sakovich equation, which belongs to second-order partial differential equations. The constructed model does not contain any dispersion term such as uxxx.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 9
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 9 August 2021

Abdul-Majid Wazwaz

This paper aims to introduce a new (3 + 1)-dimensional fourth-order integrable equation characterized by second-order derivative in time t. The new equation models both…

Abstract

Purpose

This paper aims to introduce a new (3 + 1)-dimensional fourth-order integrable equation characterized by second-order derivative in time t. The new equation models both right- and left-going waves in a like manner to the Boussinesq equation.

Design/methodology/approach

This formally uses the simplified Hirota’s method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space.

Findings

This paper confirms the complete integrability of the newly developed (3 + 1)-dimensional model in the Painevé sense.

Research limitations/implications

This paper addresses the integrability features of this model via using the Painlevé analysis.

Practical implications

This paper presents a variety of lump solutions via using a variety of numerical values of the included parameters.

Social implications

This work formally furnishes useful algorithms for extending integrable equations and for the determination of lump solutions.

Originality/value

To the best of the author’s knowledge, this paper introduces an original work with newly developed integrable equation and shows useful findings of solitons and lump solutions.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 8 January 2020

Abdul-Majid Wazwaz

The purpose of this paper is to introduce two new (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equations, the first with constant coefficients and the other with…

Abstract

Purpose

The purpose of this paper is to introduce two new (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equations, the first with constant coefficients and the other with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for the two developed models.

Design/methodology/approach

The newly developed models with constant coefficients and with time-dependent coefficients have been handled by using the simplified Hirota’s method. The author also uses the complex Hirota’s criteria for deriving multiple complex soliton solutions.

Findings

The two developed BLMP models exhibit complete integrability for any constant coefficient and any analytic time-dependent coefficients by investigating the compatibility conditions for each developed model.

Research limitations/implications

The paper presents an efficient algorithm for handling integrable equations with constant and analytic time-dependent coefficients.

Practical implications

The paper presents two new integrable equations with a variety of coefficients. The author showed that integrable equations with constant and time-dependent coefficients give real and complex soliton solutions.

Social implications

The paper presents useful algorithms for finding and studying integrable equations with constant and time-dependent coefficients.

Originality/value

The paper presents an original work with a variety of useful findings.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 9
Type: Research Article
ISSN: 0961-5539

Keywords

1 – 10 of 48