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Article
Publication date: 7 April 2015

Hanlin Chen, Zhenhui Xu and Zhengde Dai

The purpose of this paper is to reveal dynamical behavior of nonlinear wave by searching for the new breather soliton and cross two-soliton solutions of the fifth-order…

Abstract

Purpose

The purpose of this paper is to reveal dynamical behavior of nonlinear wave by searching for the new breather soliton and cross two-soliton solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG) equation.

Design/methodology/approach

The authors apply bilinear form and extended homoclinic test approach to the fifth-order CDG equation.

Findings

In this paper, by using bilinear form and extended homoclinic test approach, the authors obtain new breather soliton and cross two-soliton solutions of the fifth-order CDG equation. It is shown that the extended homoclinic test approach, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Research limitations/implications

The research manifests that the structures of the solution to nonlinear equations are diversified and complicated.

Originality/value

The methods used in this paper can be widely applied to the research of spatial and temporal characteristics of nonlinear equations in physics and engineering technology. These methods are also conducive for people to know objective laws and grasp the essential features of the development of the world.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 27 November 2018

Lakhveer Kaur and Abdul-Majid Wazwaz

The purpose of this paper is to explore new reduced form of the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation by considering its bilinear equations…

Abstract

Purpose

The purpose of this paper is to explore new reduced form of the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation by considering its bilinear equations, derived from connection between the Hirota’s transformation and Bell polynomials.

Design/methodology/approach

Based on the bilinear form of new reduced form of the (3 + 1)-dimensional generalized BKP equation, lump solutions with sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions are discovered. Also, extended homoclinic approach is applied to considered equation for finding solitary wave solutions.

Findings

A class of the bright-dark lump waves are fabricated for studying different attributes of (3 + 1)-dimensional generalized BKP equation and some new exact solutions including kinky periodic solitary wave solutions and line breathers periodic are also obtained by Following the extended homoclinic approach.

Research limitations/implications

The paper presents that the implemented methods have emerged as a promising and robust mathematical tool to manage (3 + 1)-dimensional generalized BKP equation by using the Hirota’s bilinear equation.

Practical implications

By considering important characteristics of lump and solitary wave solutions, one can understand the shapes, amplitudes and velocities of solitons after the collision with another soliton.

Social implications

The analysis of these higher-dimensional nonlinear wave equations is not only of fundamental interest but also has important practical implications in many areas of mathematical physics and ocean engineering.

Originality/value

To the best of the authors’ knowledge, the acquired solutions given in various cases have not been reported for new reduced form of the (3 + 1)-dimensional generalized BKP equation in the literature. These obtained solutions are advantageous for researchers to know objective laws and grab the indispensable features of the development of the mathematical physics.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 19 December 2018

Hui Wang and Tian-Tian Zhang

The purpose of this paper is to study stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order terms, which…

Abstract

Purpose

The purpose of this paper is to study stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order terms, which can be used to describe the propagation properties of optical soliton solutions.

Design/methodology/approach

The authors apply the ansatz method and the Hamiltonian system technique to find its bright, dark and Gaussian wave solitons and analyze its modulation instability analysis and stability analysis solution.

Findings

The results imply that the generalized nonlinear Schrödinger equation has bright, dark and Gaussian wave solitons. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior. Some constraint conditions are provided which can guarantee the existence of solitons. The authors analyze its modulation instability analysis and stability analysis solution.

Originality/value

These results may help us to further study the local structure and the interaction of solutions in generalized nonlinear Schrödinger -type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of the generalized nonlinear Schrödinger--type equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 February 2021

Shou-Fu Tian, Xiao-Fei Wang, Tian-Tian Zhang and Wang-Hua Qiu

The purpose of this paper is to study the stability analysis and optical solitary wave solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation, which are derived from a…

Abstract

Purpose

The purpose of this paper is to study the stability analysis and optical solitary wave solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation, which are derived from a multicomponent plasma with nonextensive distribution.

Design Methodology Approach

Based on the ansatz and sub-equation theories, the authors use a direct method to find stability analysis and optical solitary wave solutions of the (2 + 1)-dimensional equation.

Findings

By considering the ansatz method, the authors successfully construct the bright and dark soliton solutions of the equation. The sub-equation method is also extended to find its complexitons solutions. Moreover, the explicit power series solution is also derived with its convergence analysis. Finally, the influences of each parameter on these solutions are discussed via graphical analysis.

Originality Value

The dynamics of these solutions are analyzed to enrich the diversity of the dynamics of high-dimensional nonlinear Schrödinger equation type nonlinear wave fields.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 5 January 2015

Wei Chen, Hanlin Chen and Zhengde Dai

The purpose of this paper is to find solutions for the (2+1)-dimensional B-type Kadomtsev-Petviashvili equation and to research the quality of B-type Kadomtsev-Petviashvili…

Abstract

Purpose

The purpose of this paper is to find solutions for the (2+1)-dimensional B-type Kadomtsev-Petviashvili equation and to research the quality of B-type Kadomtsev-Petviashvili equation.

Design/methodology/approach

The authors apply the extended three-wave approach and the homoclinic test technique to solve the B-type Kadomtsev-Petviashvili equation.

Findings

The authors obtain breather type of cross-kink solutions, doubly breather type of kink solitary solutions and the breather type of kink wave solutions for B-type Kadomtsev-Petviashvili equation.

Research limitations/implications

As nonlinear evolution equations are characterized by rich dynamical behaviors, the authors have just found some of them and others are still to be found.

Originality/value

These results may help us to investigate the local structure and the interaction of waves in high-dimensional models.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 5 January 2015

Zhenhui Xu and Hanlin Chen

– The purpose of this paper is to reveal the dynamical behavior of higher dimensional nonlinear wave by searching for the multi-wave solutions to the (3+1)-D Jimbo-Miwa equation.

Abstract

Purpose

The purpose of this paper is to reveal the dynamical behavior of higher dimensional nonlinear wave by searching for the multi-wave solutions to the (3+1)-D Jimbo-Miwa equation.

Design/methodology/approach

The authors apply bilinear form and extended homoclinic test approach to the (3+1)-D Jimbo-Miwa equation.

Findings

In this paper, by using bilinear form and extended homoclinic test approach, the authors obtain new cross-kink multi-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation, including the periodic breathertype of kink three-soliton solutions, the cross-kink four-soliton solutions, the doubly periodic breather-type of soliton solutions and the doubly periodic breather-type of cross-kink two-soliton solutions. It is shown that the extended homoclinic test approach, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving higher dimensional nonlinear evolution equations in mathematical physics.

Research limitations/implications

The research manifests that the structures of the solution to higher dimensional nonlinear equations are diversified and complicated.

Originality/value

The methods used in this paper can be widely applied to the research of spatial and temporal characteristics of nonlinear equations in physics and engineering technology. These methods are also conducive for people to know objective laws and grasp the essential features of the development of the world.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 27 May 2020

Ji-Huan He, Fei-Yu Ji and Hamid Mohammad-Sedighi

The purpose of this paper is to demonstrate that the numerical method is not everything for nonlinear equations. Some properties cannot be revealed numerically; an example is used…

Abstract

Purpose

The purpose of this paper is to demonstrate that the numerical method is not everything for nonlinear equations. Some properties cannot be revealed numerically; an example is used to elucidate the fact.

Design/methodology/approach

A variational principle is established for the generalized KdV – Burgers equation by the semi-inverse method, and the equation is solved analytically by the exp-function method, and some exact solutions are obtained, including blowup solutions and discontinuous solutions. The solution morphologies are studied by illustrations using different scales.

Findings

Solitary solution is the basic property of nonlinear wave equations. This paper finds some new properties of the KdV–Burgers equation, which have not been reported in open literature and cannot be effectively elucidated by numerical methods. When the solitary solution or the blowup solution is observed on a much small scale, their discontinuous property is first found.

Originality/value

The variational principle can explain the blowup and discontinuous properties of a nonlinear wave equation, and the exp-function method is a good candidate to reveal the solution properties.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 14 September 2023

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

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 33 no. 12
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 28 March 2023

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

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 33 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 25 June 2019

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

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 11
Type: Research Article
ISSN: 0961-5539

Keywords

11 – 20 of 22