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1 – 7 of 7The purpose of this paper is concerned with developing new integrable Vakhnenko–Parkes equations with time-dependent coefficients. The author obtains multiple soliton solutions and…
Abstract
Purpose
The purpose of this paper is concerned with developing new integrable Vakhnenko–Parkes 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 Vakhnenko–Parkes 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 Vakhnenko–Parkes equations, which give a variety of multiple real and complex soliton solutions.
Details
Keywords
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…
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
Keywords
The purpose of this paper is concerned with investigating three integrable shallow water waves equations with time-dependent coefficients. The author obtains multiple soliton…
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
Keywords
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.
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Keywords
The purpose of this paper is to investigate a variety of Painlevé integrable equations derived from a Hamiltonian equation.
Abstract
Purpose
The purpose of this paper is to investigate a variety of Painlevé integrable equations derived from a Hamiltonian equation.
Design/methodology/approach
The newly developed Painlevé integrable equations have been handled by using Hirota’s direct method. The authors obtain multiple soliton solutions and other kinds of solutions for these six models.
Findings
The developed Hamiltonian models exhibit complete integrability in analogy with the original equation.
Research limitations/implications
The present study is to address these two main motivations: the study of the integrability features and solitons and other useful solutions for the developed equations.
Practical implications
The work introduces six Painlevé-integrable equations developed from a Hamiltonian 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 equations and shows useful findings.
Details
Keywords
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
Keywords
The purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.
Abstract
Purpose
The purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.
Design/methodology/approach
By means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions.
Findings
The breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before.
Originality/value
The solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.
Details