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Article
Publication date: 11 October 2018

Lian-Li Feng and Tian-Tian Zhang

The purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation…

Abstract

Purpose

The purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which can be used to describe the propagation of weakly nonlinear dispersive long waves on the surface of a fluid.

Design/methodology/approach

The authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional gKP equation.

Findings

The results imply that the gKP equation admits rogue waves, homoclinic breather waves and soliton waves. Moreover, the authors also find that rogue waves can come from the extreme behavior of the breather solitary wave. The authors analyze the propagation and interaction properties of these solutions to better understand the dynamic behavior of these solutions.

Originality/value

These results may help us to further study the local structure and the interaction of waves in KP-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

Details

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

Keywords

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Article
Publication date: 10 July 2019

Hui Wang, Shou-Fu Tian and Yi Chen

The purpose of this paper is to study the breather waves, rogue waves and solitary waves of an extended (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation, which can…

Abstract

Purpose

The purpose of this paper is to study the breather waves, rogue waves and solitary waves of an extended (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation, which can be used to depict many nonlinear phenomena in fluid dynamics and plasma physics.

Design/methodology/approach

The authors apply the Bell’s polynomial approach, the homoclinic test technique and Hirota’s bilinear method to find the breather waves, rogue waves and solitary waves of the extended (3 + 1)-dimensional KP equation.

Findings

The results imply that the extended (3 + 1)-dimensional KP equation has breather wave, rogue wave and solitary wave solutions. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior.

Originality/value

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

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 27 November 2018

Jin-Jin Mao, Shou-Fu Tian and Tian-Tian Zhang

The purpose of this paper is to find the exact solutions of a (3 + 1)-dimensional non-integrable Korteweg-de Vries type (KdV-type) equation, which can be used to describe…

Abstract

Purpose

The purpose of this paper is to find the exact solutions of a (3 + 1)-dimensional non-integrable Korteweg-de Vries type (KdV-type) equation, which can be used to describe the stability of soliton in a nonlinear media with weak dispersion.

Design/methodology/approach

The authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional non-integrable KdV-type equation. The used approach formally derives the essential conditions for these solutions to exist.

Findings

The results show that the equation exists rogue waves, homoclinic breather waves and soliton waves. To better understand the dynamic behavior of these solutions, the authors analyze the propagation and interaction properties of the these solutions.

Originality/value

These results may help to investigate the local structure and the interaction of waves in KdV-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

Details

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

Keywords

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Article
Publication date: 3 December 2020

Na Liu

The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an…

Abstract

Purpose

The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an interaction of long waves with short wave packets.

Design/methodology/approach

The author applies the Hirota’s bilinear method, extended homoclinic test approach and parameter limit method to construct the homoclinic breather waves and rogue waves of the (2 + 1)-dimensional Mel’nikov equation. Moreover, multi-soliton waves are constructed by using the three-wave method.

Findings

The results imply that the (2 + 1)-dimensional Mel’nikov equation has breather waves, rogue waves and multi-soliton waves. Moreover, the dynamic properties of such solutions are displayed vividly by figures.

Research limitations/implications

This paper presents efficient methods to find breather waves, rogue waves and multi-soliton waves for nonlinear evolution equations.

Originality/value

The outcome suggests that the extreme behavior of the homoclinic breather waves yields the rogue waves. Moreover, the multi-soliton waves are constructed, including the new breather two-solitary and two-soliton solutions. Meanwhile, the dynamics of these solutions will greatly enrich the diversity of the dynamics of the (2 + 1)-dimensional Mel’nikov equation.

Details

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

Keywords

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Article
Publication date: 6 August 2019

Jin-Jin Mao, Shou-Fu Tian, Xing-Jie Yan and Tian-Tian Zhang

The purpose of this study is to examine the lump solutions of the (3 + 1)-dimensional nonlinear evolution equations by considering a (3 + 1)-dimensional generalized…

Abstract

Purpose

The purpose of this study is to examine the lump solutions of the (3 + 1)-dimensional nonlinear evolution equations by considering a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation and a (3 + 1)-dimensional variable-coefficient generalized B-type Kadomtsev–Petviashvili (vcgBKP) equation as examples.

Design/methodology/approach

Based on Hirota’s bilinear theory, a direct method is used to examine the lump solutions of these two equations.

Findings

The complete non-elastic interaction solutions between a lump and a stripe are also discussed for the equations, which show that the lump solitons are swallowed by the stripe solitons.

Originality/value

The dynamics of these solutions are analyzed to enrich the diversity of the dynamics of high-dimensional KP-type nonlinear wave equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 9
Type: Research Article
ISSN: 0961-5539

Keywords

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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…

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

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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…

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

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Article
Publication date: 7 April 2015

Changfu Liu, Zeping Wang, Zhengde Dai and Longwei Chen

The purpose of this paper is to construct analytical solutions of the (2+1)-dimensional nonlinear Schrodinger equations, and the existence of rogue waves and their…

Abstract

Purpose

The purpose of this paper is to construct analytical solutions of the (2+1)-dimensional nonlinear Schrodinger equations, and the existence of rogue waves and their localized structures are studied.

Design/methodology/approach

Function transformation and variable separation method are applied to the (2+1)-dimensional nonlinear Schrodinger equations.

Findings

A series of analytical solutions including rogue wave solutions for the (2+1)-dimensional nonlinear Schrodinger equations are constructed. Localized structures of rogue waves confirm the presence of large amplitude wave wall.

Research limitations/implications

The localized structures of rogue waves are displayed by analytical solutions and figures. The authors just find some of them and others still to be found.

Originality/value

These results may help to investigate the localized structures and the behavior of rogue waves for nonlinear evolution equations. Applying two different function transformations and variable separation functions to two different states of the equations, respectively, to construct the solutions of the (2+1)-dimensional nonlinear Schrodinger equations. Rogue wave solutions are enumerated and their figures are partly displayed.

Details

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

Keywords

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Article
Publication date: 25 June 2019

Abdul-Majid Wazwaz

The purpose of this paper is concerned with developing new integrable Vakhnenko–Parkes equations with time-dependent coefficients. The author obtains multiple soliton…

Abstract

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.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 12
Type: Research Article
ISSN: 0961-5539

Keywords

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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…

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

1 – 10 of 290