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 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.

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.

Painlevé analysis for three integrable shallow water waves equations with time-dependent coefficients

2019 ◽  
Vol 30 (2) ◽  
pp. 996-1008 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Direct Method ◽  
Soliton Solutions ◽  
Handling Time ◽  
Time Dependent ◽  
Shallow Water Waves ◽  
Integrable Equations ◽  
Content Type ◽  
Multiple Complex Soliton Solutions

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.

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.

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.

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.

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.

An extended time-dependent KdV6 equation

2019 ◽  
Vol 29 (11) ◽  
pp. 4205-4212
Author(s):  
Abdul-Majid Wazwaz ◽  
Gui-Qiong Xu
Keyword(s):  
Direct Method ◽  
Soliton Solutions ◽  
Complete Integrability ◽  
Time Dependent ◽  
Content Type ◽  
Dependent Model ◽  
First Time ◽  
Multiple Complex Soliton Solutions ◽  
Kdv6 Equation ◽  
New Time

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.

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