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Article
Publication date: 25 June 2019

Abdul-Majid Wazwaz

The purpose of this paper is concerned with developing new integrable VakhnenkoParkes equations with time-dependent coefficients. The author obtains multiple soliton…

Abstract

Purpose

The purpose of this paper is concerned with developing new integrable VakhnenkoParkes equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for the time-dependent equations.

Design/methodology/approach

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

Findings

The developed integrable models exhibit complete integrability for any analytic time-dependent coefficient.

Research limitations/implications

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

Practical implications

The author develops two VakhnenkoParkes equations with time-dependent coefficients. These models represent more specific data than the related equations with constant coefficients. The author showed that integrable equations with time-dependent coefficients give real and complex soliton solutions.

Social implications

The work presents useful techniques for finding integrable equations with time-dependent coefficients.

Originality/value

The paper gives new integrable VakhnenkoParkes equations, which give a variety of multiple real and complex soliton solutions.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 12
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 VakhnenkoParkes equation. The author formally derives multiple soliton solutions for…

Abstract

Purpose

This study aims to develop a new (3 + 1)-dimensional Painlevé-integrable extended VakhnenkoParkes 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 VakhnenkoParkes model exhibits complete integrability in analogy with the standard VakhnenkoParkes 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: 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: 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: 3 December 2020

Na Liu

The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an…

Abstract

Purpose

The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an interaction of long waves with short wave packets.

Design/methodology/approach

The author applies the Hirota’s bilinear method, extended homoclinic test approach and parameter limit method to construct the homoclinic breather waves and rogue waves of the (2 + 1)-dimensional Mel’nikov equation. Moreover, multi-soliton waves are constructed by using the three-wave method.

Findings

The results imply that the (2 + 1)-dimensional Mel’nikov equation has breather waves, rogue waves and multi-soliton waves. Moreover, the dynamic properties of such solutions are displayed vividly by figures.

Research limitations/implications

This paper presents efficient methods to find breather waves, rogue waves and multi-soliton waves for nonlinear evolution equations.

Originality/value

The outcome suggests that the extreme behavior of the homoclinic breather waves yields the rogue waves. Moreover, the multi-soliton waves are constructed, including the new breather two-solitary and two-soliton solutions. Meanwhile, the dynamics of these solutions will greatly enrich the diversity of the dynamics of the (2 + 1)-dimensional Mel’nikov equation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

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