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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 Kadomtsev

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 KadomtsevPetviashvili (gKP) equation and a (3 + 1)-dimensional variable-coefficient generalized B-type KadomtsevPetviashvili (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

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: 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 KadomtsevPetviashvili (KP) equation, which can be used…

Abstract

Purpose

The purpose of this paper is to study the breather waves, rogue waves and solitary waves of an extended (3 + 1)-dimensional KadomtsevPetviashvili (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

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 KadomtsevPetviashvili (gKP) equation, which…

Abstract

Purpose

The purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized KadomtsevPetviashvili (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

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 KadomtsevPetviashvili equation characterized by adding three new linear terms.

Abstract

Purpose

This study aims to propose an extended (3 + 1)-dimensional integrable KadomtsevPetviashvili 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: 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

Open Access
Article
Publication date: 14 March 2023

Svetlin Georgiev, Aissa Boukarou, Keltoum Bouhali and Khaled Zennir

This paper is devoted to the generalized KadomtsevPetviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global…

Abstract

Purpose

This paper is devoted to the generalized KadomtsevPetviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Design/methodology/approach

This paper is devoted to the generalized KadomtsevPetviashviliequation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Findings

This paper is devoted to the generalized KadomtsevPetviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Originality/value

This article is devoted to the generalized KadomtsevPetviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Details

Arab Journal of Mathematical Sciences, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 1319-5166

Keywords

Article
Publication date: 9 August 2021

Abdul-Majid Wazwaz

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…

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.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 7 June 2023

Na Liu

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…

40

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

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 33 no. 9
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 25 October 2021

Liu-Qing Li, Yi-Tian Gao, Xin Yu, Gao-Fu Deng and Cui-Cui Ding

This paper aims to study the Gramian solutions and solitonic interactions of a (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, which models the nonlinear and dispersive…

Abstract

Purpose

This paper aims to study the Gramian solutions and solitonic interactions of a (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, which models the nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth.

Design/methodology/approach

Pfaffian technique is used to construct the Gramian solutions of the (2 + 1)-dimensional BKK system. Asymptotic analysis is applied on the two-soliton solutions to study the interaction properties.

Findings

N-soliton solutions in the Gramian with a real function ζ(y) of the (2 + 1)-dimensional BKK system are constructed and proved, where N is a positive integer and y is the scaled space variable. Conditions of elastic and inelastic interactions between the two solitons are revealed asymptotically. For the three and four solitons, elastic, inelastic interactions and soliton resonances are discussed graphically. Effect of the wave numbers, initial phases and ζ(y) on the solitonic interactions is also studied.

Originality/value

Shallow water waves are studied for the applications in environmental engineering and hydraulic engineering. This paper studies the shallow water waves through the Gramian solutions of a (2 + 1)-dimensional BKK system and provides some phenomena that have not been studied.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

1 – 10 of 32