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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, 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
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
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
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, 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
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
Kang-Jia Wang, Guo-Dong Wang and Feng Shi
The fractal and fractional calculus have obtained considerable attention in the electrical and electronic engineering since they can model many complex phenomena that the…
Abstract
Purpose
The fractal and fractional calculus have obtained considerable attention in the electrical and electronic engineering since they can model many complex phenomena that the traditional integer-order calculus cannot. The purpose of this paper is to develop a new fractional pulse narrowing nonlinear transmission lines model within the local fractional calculus for the first time and derive a novel method, namely, the direct mapping method, to seek for the nondifferentiable (ND) exact solutions.
Design/methodology/approach
By defining some special functions via the Mittag–Leffler function on the Cantor sets, a novel approach, namely, the direct mapping method is derived via constructing a group of the nonlinear local fractional ordinary differential equations. With the aid of the direct mapping method, four groups of the ND exact solutions are obtained in just one step. The dynamic behaviors of the ND exact solutions on the Cantor sets are also described through the 3D graphical illustration.
Findings
It is found that the proposed method is simple but effective and can construct four sets of the ND exact solutions in just one step. In addition, one of the ND exact solutions becomes the exact solution of the classic pulse narrowing nonlinear transmission lines model for the special case 9 = 1, which strongly proves the correctness and effectiveness of the method. The ideas in the paper can be used to study the other fractal partial differential equations (PDEs) within the local fractional derivative (LFD) arising in electrical and electronic engineering.
Originality/value
The fractional pulse narrowing nonlinear transmission lines model within the LFD is proposed for the first time in this paper. The proposed method in the work can be used to study the other fractal PDEs arising in electrical and electronic engineering. The findings in this work are expected to shed a light on the study of the fractal PDEs arising in electrical and electronic engineering.
Details