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.
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Abdul-Majid Wazwaz, Weaam Alhejaili and Samir El-Tantawy
This study aims to explore a novel model that integrates the Kairat-II equation and Kairat-X equation (K-XE), denoted as the Kairat-II-X (K-II-X) equation. This model demonstrates…
Abstract
Purpose
This study aims to explore a novel model that integrates the Kairat-II equation and Kairat-X equation (K-XE), denoted as the Kairat-II-X (K-II-X) equation. This model demonstrates the connections between the differential geometry of curves and the concept of equivalence.
Design/methodology/approach
The Painlevé analysis shows that the combined K-II-X equation retains the complete Painlevé integrability.
Findings
This study explores multiple soliton (solutions in the form of kink solutions with entirely new dispersion relations and phase shifts.
Research limitations/implications
Hirota’s bilinear technique is used to provide these novel solutions.
Practical implications
This study also provides a diverse range of solutions for the K-II-X equation, including kink, periodic and singular solutions.
Social implications
This study provides formal procedures for analyzing recently developed systems that investigate optical communications, plasma physics, oceans and seas, fluid mechanics and the differential geometry of curves, among other topics.
Originality/value
The study introduces a novel Painlevé integrable model that has been constructed and delivers valuable discoveries.
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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.
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Abdul-Majid Wazwaz, Wedad Albalawi and Samir A. El-Tantawy
The purpose of this paper is to study an extended hierarchy of nonlinear evolution equations including the sixth-order dispersion Korteweg–de Vries (KdV6), eighth-order dispersion…
Abstract
Purpose
The purpose of this paper is to study an extended hierarchy of nonlinear evolution equations including the sixth-order dispersion Korteweg–de Vries (KdV6), eighth-order dispersion KdV (KdV8) and many other related equations.
Design/methodology/approach
The newly developed models have been handled using the simplified Hirota’s method, whereas multiple soliton solutions are furnished using Hirota’s criteria.
Findings
The authors show that every member of this hierarchy is characterized by distinct dispersion relation and distinct resonance branches, whereas the phase shift retains the KdV type of shifts for any member.
Research limitations/implications
This paper presents an efficient algorithm for handling a hierarchy of integrable equations of diverse orders.
Practical implications
Multisoliton solutions are derived for each member of the hierarchy, and then generalized for any higher-order model.
Social implications
This work presents useful algorithms for finding and studying integrable equations of a hierarchy of nonlinear equations. The developed models exhibit complete integrability, by investigating the compatibility conditions for each model.
Originality/value
This paper presents an original work with a variety of useful findings.
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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.
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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.