A (2 + 1)-dimensional extension of the Benjamin-Ono equation

2018 ◽  
Vol 28 (11) ◽  
pp. 2681-2687 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Content Type ◽  
Multiple Soliton Solutions ◽  
Proposed Model ◽  
New Findings ◽  
Propagation Of Waves ◽  
Model Features ◽  
Multiple Complex Soliton Solutions ◽  
Practical Implications

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.

A variety of completely integrable Calogero–Bogoyavlenskii–Schiff equations with time-dependent coefficients

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Time Dependent ◽  
Integrable Equations ◽  
Content Type ◽  
Completely Integrable ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Propagation Of Waves ◽  
Multiple Complex Soliton Solutions ◽  
Practical Implications

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.

Two new Painlevé-integrable extended Sakovich equations with (2 + 1) and (3 + 1) dimensions

2019 ◽  
Vol 30 (3) ◽  
pp. 1379-1387 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Design Methodology ◽  
Original Work ◽  
Direct Method ◽  
Soliton Solutions ◽  
Complete Integrability ◽  
Integrable Equations ◽  
Content Type ◽  
Multiple Soliton Solutions ◽  
Multiple Complex Soliton Solutions ◽  
Practical Implications

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.

Painlevé analysis for new (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equations with constant and time-dependent coefficients

2019 ◽  
Vol 30 (9) ◽  
pp. 4259-4266 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Design Methodology ◽  
Soliton Solutions ◽  
Time Dependent ◽  
Integrable Equations ◽  
Content Type ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Constant Coefficients ◽  
Multiple Complex Soliton Solutions ◽  
Practical Implications

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.

New (3+1)-dimensional integrable fourth-order nonlinear equation: lumps and multiple soliton solutions

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Fourth Order ◽  
Soliton Solutions ◽  
Integrable Equation ◽  
Lump Solutions ◽  
Content Type ◽  
Multiple Soliton Solutions ◽  
Order Nonlinear Equation ◽  
New Equation ◽  
Practical Implications

Purpose This paper aims to introduce a new (3 + 1)-dimensional fourth-order integrable equation characterized by second-order derivative in time t. The new equation models both right- and left-going waves in a like manner to the Boussinesq equation. Design/methodology/approach This formally uses 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 This paper confirms the complete integrability of the newly developed (3 + 1)-dimensional model in the Painevé sense. Research limitations/implications This paper addresses the integrability features of this model via using the Painlevé analysis. Practical implications This paper presents a variety of lump solutions via using a variety of numerical values of the included parameters. Social implications This work formally furnishes useful algorithms for extending integrable equations and for the determination of lump solutions. Originality/value To the best of the author’s knowledge, this paper introduces an original work with newly developed integrable equation and shows useful findings of solitons and lump solutions.

Two integrable third-order and fifth-order KdV equations with time-dependent coefficients

2019 ◽  
Vol 29 (6) ◽  
pp. 2093-2102 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Efficient Algorithm ◽  
Soliton Solutions ◽  
Integrable Equation ◽  
Time Dependent ◽  
Integrable Equations ◽  
Content Type ◽  
Kdv Equations ◽  
Propagation Of Waves ◽  
Fifth Order ◽  
Multiple Complex Soliton Solutions

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.

A new (3 + 1)-dimensional Painlevé-integrable Sakovich equation: multiple soliton solutions

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Design Methodology ◽  
Soliton Solutions ◽  
Complete Integrability ◽  
Content Type ◽  
Hirota’S Method ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Hirota's Method ◽  
Practical Implications ◽  
Dispersion Term

Purpose This paper aims to develop a new (3 + 1)-dimensional Painlev´e-integrable extended Sakovich equation. This paper formally derives multiple soliton solutions for this developed model. Design/methodology/approach This paper uses the simplified Hirota’s method for deriving multiple soliton solutions. Findings This paper finds that the developed (3 + 1)-dimensional Sakovich model exhibits complete integrability in analogy with the standard Sakovich equation. Research limitations/implications This paper addresses the integrability features of this model via using the Painlev´e analysis. This paper reports multiple soliton solutions for this equation by using the simplified Hirota’s method. Practical implications The study reports three non-linear terms added to the standard Sakovich equation. Social implications The study presents useful algorithms for constructing new integrable equations and for handling these equations. Originality/value The paper reports a new Painlev´e-integrable extended Sakovich equation, which belongs to second-order partial differential equations. The constructed model does not contain any dispersion term such as uxxx.

Two wave mode higher-order modified KdV equations

2017 ◽  
Vol 27 (10) ◽  
pp. 2223-2230 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Wave Model ◽  
Wave Mode ◽  
Higher Order ◽  
Dispersion Parameters ◽  
Content Type ◽  
Kdv Equations ◽  
Multiple Soliton Solutions ◽  
Modified Kdv Equations ◽  
Propagation Of Waves

Purpose The purpose of this paper is concerned with developing two-mode higher-order modified Korteweg-de Vries (KdV) equations. The study shows that multiple soliton solutions exist for essential conditions related to the nonlinearity and dispersion parameters. Design/methodology/approach The proposed technique for constructing a two-wave model, as presented in this work, has been shown to be very efficient. The employed approach formally derives the essential conditions for soliton solutions to exist. Findings The examined two-wave model features interesting results in propagation of waves and fluid flow. Research limitations/implications The paper presents a new and efficient algorithm for constructing and studying two-wave-mode higher-order modified KdV equations. Practical implications A two-wave model was constructed for higher-order modified KdV equations. The essential conditions for multiple soliton solutions to exist were derived. Social implications The work shows the distinct features of the standard equation and the newly developed equation. Originality/value The work is original and this is the first time for two-wave-mode higher-order modified KdV equations to be constructed and studied.

Higher dimensional integrable Vakhnenko–Parkes equation: multiple soliton solutions

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Design Methodology ◽  
Soliton Solutions ◽  
Integrable Equation ◽  
Content Type ◽  
Hirota’S Method ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Higher Dimensional ◽  
Hirota's Method ◽  
Practical Implications

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.

New integrable Vakhnenko–Parkes (VP) equations with time-dependent coefficients

2019 ◽  
Vol 29 (12) ◽  
pp. 4598-4606 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Direct Method ◽  
Soliton Solutions ◽  
Handling Time ◽  
Time Dependent ◽  
Specific Data ◽  
Integrable Equations ◽  
Content Type ◽  
Constant Coefficients ◽  
Multiple Complex Soliton Solutions ◽  
Practical Implications

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.

New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Lump Solutions ◽  
Content Type ◽  
Shallow Water Wave ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Shallow Water Wave Equations

Purpose This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space. Design/methodology/approach The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense. Findings The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium. Research limitations/implications The author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method. Practical implications The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition. Social implications The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions. Originality/value The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

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