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1 – 10 of 186The purpose of this paper is concerned with developing two integrable Korteweg de-Vries (KdV) equations of third- and fifth-orders; each possesses time-dependent coefficients. The…
Abstract
Purpose
The purpose of this paper is concerned with developing two integrable Korteweg de-Vries (KdV) equations of third- and fifth-orders; each possesses time-dependent coefficients. The study shows that multiple soliton solutions exist and multiple complex soliton solutions exist for these two equations.
Design/methodology/approach
The integrability of each of the developed models has been confirmed by using the Painlev´e analysis. The author uses the complex forms of the simplified Hirota’s method to obtain two fundamentally different sets of solutions, multiple real and multiple complex soliton solutions for each model.
Findings
The time-dependent KdV equations feature interesting results in propagation of waves and fluid flow.
Research limitations/implications
The paper presents a new efficient algorithm for constructing time-dependent integrable equations.
Practical implications
The author develops two time-dependent integrable KdV equations of third- and fifth-order. These models represent more specific data than the constant equations. The author showed that integrable equation gives real and complex soliton solutions.
Social implications
The work presents useful findings in the propagation of waves.
Originality/value
The paper presents a new efficient algorithm for constructing time-dependent integrable equations.
Details
Keywords
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…
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
The purpose of this paper is concerned with developing a (2 + 1)-dimensional Benjamin–Ono equation. The study shows that multiple soliton solutions exist and multiple complex…
Abstract
Purpose
The purpose of this paper is concerned with developing a (2 + 1)-dimensional Benjamin–Ono equation. The study shows that multiple soliton solutions exist and multiple complex soliton solutions exist for this equation.
Design/methodology/approach
The proposed model has been handled by using the Hirota’s method. Other techniques were used to obtain traveling wave solutions.
Findings
The examined extension of the Benjamin–Ono model features interesting results in propagation of waves and fluid flow.
Research limitations/implications
The paper presents a new efficient algorithm for constructing extended models which give a variety of multiple soliton solutions.
Practical implications
This work is entirely new and provides new findings, where although the new model gives multiple soliton solutions, it is nonintegrable.
Originality/value
The work develops two complete sets of multiple soliton solutions, the first set is real solitons, whereas the second set is complex solitons.
Details
Keywords
The purpose of this paper is to introduce a variety of new completely integrable Calogero–Bogoyavlenskii–Schiff (CBS) equations with time-dependent coefficients. The author…
Abstract
Purpose
The purpose of this paper is to introduce a variety of new completely integrable Calogero–Bogoyavlenskii–Schiff (CBS) equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for each of the developed models.
Design/methodology/approach
The newly developed models with time-dependent coefficients have been handled by using the simplified Hirota’s method. Moreover, multiple complex soliton solutions are derived by using complex Hirota’s criteria.
Findings
The developed models exhibit complete integrability, for specific determined functions, by investigating the compatibility conditions for each model.
Research limitations/implications
The paper presents an efficient algorithm for handling integrable equations with analytic time-dependent coefficients.
Practical implications
The work presents new integrable equations with a variety of time-dependent coefficients. The author showed that integrable equations with time-dependent coefficients give real and complex soliton solutions.
Social implications
This study presents useful algorithms for finding and studying integrable equations with time-dependent coefficients.
Originality/value
The paper gives new integrable CBS equations which appear in propagation of waves and provide a variety of multiple real and complex soliton solutions.
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 solutions…
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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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
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 (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
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
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