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1 – 10 of 28This 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…
Abstract
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.
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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.
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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.
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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.
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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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The purpose of this paper is to concern with a reliable treatment of the (2+1)-dimensional and the (3+1)-dimensional logarithmic Boussinesq equations (BEs). The author uses the…
Abstract
Purpose
The purpose of this paper is to concern with a reliable treatment of the (2+1)-dimensional and the (3+1)-dimensional logarithmic Boussinesq equations (BEs). The author uses the sense of the Gaussian solitary waves to determine these gaussons. The study confirms that models characterized by logarithmic nonlinearity give gaussons solitons of distinct physical structures.
Design/methodology/approach
The proposed technique, as presented in this work has been shown to be very efficient for solving nonlinear equations with logarithmic nonlinearity.
Findings
The (2+1) and the (3+1)-dimensional BEs were examined as well. The examined models feature interesting results in propagation of waves and fluid flow.
Research limitations/implications
The paper presents a new efficient algorithm for the higher dimensional logarithmic BEs.
Practical implications
The work shows the effect of logarithmic nonlinearity compared to other nonlinearities where standard solitons appear in the last case.
Social implications
The work will benefit audience who are willing to examine the effect of logarithmic nonlinearity.
Originality/value
The paper presents a new efficient algorithm for the higher dimensional logarithmic BEs.
Details
Keywords
Camilo Andrés Bayona Roa, Joan Baiges and R Codina
The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation…
Abstract
Purpose
The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated.
Design/methodology/approach
The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion.
Findings
Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state.
Originality/value
A complete investigation of the stabilized formulation of the compressible problem is addressed.
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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
This study aims to introduce a variety of integrable Boussinesq equations with distinct dimensions.
Abstract
Purpose
This study aims to introduce a variety of integrable Boussinesq equations with distinct dimensions.
Design/methodology/approach
The author formally uses the simplified Hirota’s method and lump schemes for exploring lump solutions, which are rationally localized in all directions in space.
Findings
The author confirms the lump solutions for every model illustrated by some graphical representations.
Research limitations/implications
The author examines the features of the obtained lumps solutions.
Practical implications
The author presents a variety of lump solutions via using a variety of numerical values of the included parameters.
Social implications
This study formally furnishes useful algorithms for using symbolic computation with Maple for the determination of lump solutions.
Originality/value
This paper introduces an original work with newly useful findings of lump solutions.
Details
Keywords
J.I. Ramos and Carmen María García López
The purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the…
Abstract
Purpose
The purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the propagation of small-amplitude waves in shallow water, as a function of the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of three types of initial conditions.
Design/methodology/approach
An implicit, first-order accurate in time, finite difference method valid for semipositive relaxation times has been used to solve the equation in a truncated domain for three different initial conditions, a first-order time derivative initially equal to zero and several constant wave speeds.
Findings
The numerical experiments show a very rapid transient from the initial conditions to the formation of a leading propagating wave, whose duration depends strongly on the shape, amplitude and width of the initial data as well as on the coefficients of the bidirectional equation. The blowup times for the triangular conditions have been found to be larger than those for the Gaussian ones, and the latter are larger than those for rectangular conditions, thus indicating that the blowup time decreases as the smoothness of the initial conditions decreases. The blowup time has also been found to decrease as the relaxation time, degree of nonlinearity, linear drift coefficient and amplitude of the initial conditions are increased, and as the width of the initial condition is decreased, but it increases as the viscosity coefficient is increased. No blowup has been observed for relaxation times smaller than one-hundredth, viscosity coefficients larger than ten-thousandths, quadratic and cubic nonlinearities, and initial Gaussian, triangular and rectangular conditions of unity amplitude.
Originality/value
The blowup of a one-dimensional, bidirectional equation that is a model for the propagation of waves in shallow water, longitudinal displacement in homogeneous viscoelastic bars, nerve conduction, nonlinear acoustics and heat transfer in very small devices and/or at very high transfer rates has been determined numerically as a function of the linear and nonlinear drift coefficients, power of the nonlinear drift, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of the initial conditions for nonzero relaxation times.
Details