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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
Abdul-Majid Wazwaz, Weaam Alhejaili and Samir El-Tantawy
The purpose of this study is to form a linear structure of components of the modified Korteweg–De Vries (mKdV) hierarchy. The new model includes 3rd order standard mKdV equation…
Abstract
Purpose
The purpose of this study is to form a linear structure of components of the modified Korteweg–De Vries (mKdV) hierarchy. The new model includes 3rd order standard mKdV equation, 5th order and 7th order mKdV equations.
Design/methodology/approach
The authors investigate Painlevé integrability of the constructed linear structure.
Findings
The Painlevé analysis demonstrates that established sum of integrable models retains the integrability of each component.
Research limitations/implications
The research also presents a set of rational schemes of trigonometric and hyperbolic functions to derive breather solutions.
Practical implications
The authors also furnish a variety of solitonic solutions and complex solutions as well.
Social implications
The work formally furnishes algorithms for extending integrable equations that consist of components of a hierarchy.
Originality/value
The paper presents an original work for developing Painlevé integrable model via using components of a hierarchy.
Details
Keywords
Abdul-Majid Wazwaz, Mansoor Alshehri and Samir A. El-Tantawy
This study aims to explore novel solitary wave solutions of a new (3 + 1)-dimensional nonlocal Boussinesq equation that illustrates nonlinear water dynamics.
Abstract
Purpose
This study aims to explore novel solitary wave solutions of a new (3 + 1)-dimensional nonlocal Boussinesq equation that illustrates nonlinear water dynamics.
Design/methodology/approach
The authors use the Painlevé analysis to study its complete integrability in the Painlevé sense.
Findings
The Painlevé analysis demonstrates the compatibility condition for the model integrability with the addition of new extra terms.
Research limitations/implications
The phase shifts, phase variables and Hirota’s bilinear algorithm are used to furnish multiple soliton solutions.
Practical implications
The authors also furnish a variety of numerous periodic solutions, kink solutions and singular solutions.
Social implications
The work formally furnishes algorithms for investigating several physical systems, including plasma physics, optical communications and oceans and seas, among others.
Originality/value
This paper presents an original work using a newly developed Painlevé integrable model, as well as novel and insightful findings.
Details
Keywords
This study aims to investigate two newly developed (3 + 1)-dimensional Kairat-II and Kairat-X equations that illustrate relations with the differential geometry of curves and…
Abstract
Purpose
This study aims to investigate two newly developed (3 + 1)-dimensional Kairat-II and Kairat-X equations that illustrate relations with the differential geometry of curves and equivalence aspects.
Design/methodology/approach
The Painlevé analysis confirms the complete integrability of both Kairat-II and Kairat-X equations.
Findings
This study explores multiple soliton solutions for the two examined models. Moreover, the author showed that only Kairat-X give lump solutions and breather wave solutions.
Research limitations/implications
The Hirota’s bilinear algorithm is used to furnish a variety of solitonic solutions with useful physical structures.
Practical implications
This study also furnishes a variety of numerous periodic solutions, kink solutions and singular solutions for Kairat-II equation. In addition, lump solutions and breather wave solutions were achieved from Kairat-X model.
Social implications
The work formally furnishes algorithms for studying newly constructed systems that examine plasma physics, optical communications, oceans and seas and the differential geometry of curves, among others.
Originality/value
This paper presents an original work that presents two newly developed Painlev\'{e} integrable models with insightful findings.
Details
Keywords
Abdul-Majid Wazwaz, Haifa A. Alyousef and Samir El-Tantawy
This study aims to propose an extended (3 + 1)-dimensional integrable Kadomtsev–Petviashvili equation characterized by adding three new linear terms.
Abstract
Purpose
This study aims to propose an extended (3 + 1)-dimensional integrable Kadomtsev–Petviashvili equation characterized by adding three new linear terms.
Design/methodology/approach
This study formally uses Painlevé test to confirm the integrability of the new system.
Findings
The Painlevé analysis shows that the compatibility condition for integrability does not die away by adding three new linear terms with distinct coefficients.
Research limitations/implications
This study uses the Hirota's bilinear method to explore multiple soliton solutions where phase shifts and phase variable are explored.
Practical implications
This study also furnishes a class of lump solutions (LSs), which are rationally localized in all directions in space, using distinct values of the parameters via using the positive quadratic function method.
Social implications
This study also shows the power of the simplified Hirota’s method in handling integrable equations.
Originality/value
This paper introduces an original work with newly developed Painlevé integrable model and shows new useful findings.
Details
Keywords
Abdul-Majid Wazwaz, Lamiaa El-Sherif and Samir El-Tantawy
This paper aims to propose a new (3+1)-dimensional integrable Hirota bilinear equation characterized by five linear partial derivatives and three nonlinear partial derivatives.
Abstract
Purpose
This paper aims to propose a new (3+1)-dimensional integrable Hirota bilinear equation characterized by five linear partial derivatives and three nonlinear partial derivatives.
Design/methodology/approach
The authors formally use 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
The Painlevé analysis shows that the compatibility condition for integrability does not die away at the highest resonance level, but integrability characteristics is justified through the Lax sense.
Research limitations/implications
Multiple-soliton solutions are explored using the Hirota's bilinear method. The authors also furnish a class of lump solutions using distinct values of the parameters via the positive quadratic function method.
Practical implications
The authors also retrieve a bunch of other solutions of distinct structures such as solitonic, periodic solutions and ratio of trigonometric functions solutions.
Social implications
This work formally furnishes algorithms for extending integrable equations and for the determination of lump solutions.
Originality/value
To the best of the authors’ knowledge, this paper introduces an original work with newly developed Lax-integrable equation and shows new useful findings.
Details
Keywords
The aim of this study is to offer a contemporary approach for getting optical soliton and traveling wave solutions for the Date–Jimbo–Kashiwara–Miwa equation.
Abstract
Purpose
The aim of this study is to offer a contemporary approach for getting optical soliton and traveling wave solutions for the Date–Jimbo–Kashiwara–Miwa equation.
Design/methodology/approach
The approach is based on a recently constructed ansätze strategy. This method is an alternative to the Painleve test analysis, producing results similarly, but in a more practical, straightforward manner.
Findings
The approach proved the existence of both singular and optical soliton solutions. The method and its application show how much better and simpler this new strategy is than current ones. The most significant benefit is that it may be used to solve a wide range of partial differential equations that are encountered in practical applications.
Originality/value
The approach has been developed recently, and this is the first time that this method is applied successfully to extract soliton solutions to the Date–Jimbo–Kashiwara–Miwa 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
Keywords
This paper aims to study the breather, lump-kink and interaction solutions of a (3 + 1)-dimensional generalized shallow water waves (GSWW) equation, which describes water waves…
Abstract
Purpose
This paper aims to study the breather, lump-kink and interaction solutions of a (3 + 1)-dimensional generalized shallow water waves (GSWW) equation, which describes water waves propagating in the ocean or is used for simulating weather.
Design/methodology/approach
Hirota bilinear form and the direct method are used to construct breather and lump-kink solutions of the GSWW equation. The “rational-cosh-cos-type” test function is applied to obtain three kinds of interaction solutions.
Findings
The fusion and fission of the interaction solutions between a lump wave and a 1-kink soliton of the GSWW equation are studied. The dynamics of three kinds of interaction solutions between lump, kink and periodic waves are discussed graphically.
Originality/value
This paper studies the breather, lump-kink and interaction solutions of the GSWW equation by using various approaches and provides some phenomena that have not been studied.
Details