Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation

Exact periodic and localized solutions of a nonlocal Mel′nikov equation are derived by the Hirota bilinear method. Many conventional nonlocal operators involve integration over a spatial or temporal domain. However, the present class of nonlocal equations depends on properties at selected far field...

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Main Authors:	Wei Liu, Zhenyun Qin, Kwok Wing Chow, Senyue Lou
Format:	Article
Language:	English
Published:	Hindawi-Wiley 2020-01-01
Series:	Complexity
Online Access:	http://dx.doi.org/10.1155/2020/2642654

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spelling	doaj-a7b8577aca664bfaba5b1f6e0faa8a112020-11-25T03:18:09ZengHindawi-WileyComplexity1076-27871099-05262020-01-01202010.1155/2020/26426542642654Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov EquationWei Liu0Zhenyun Qin1Kwok Wing Chow2Senyue Lou3College of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, ChinaSchool of Mathematics and Key Laboratory for Nonlinear Mathematical Models and Methods, Fudan University, Shanghai 200433, ChinaDepartment of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong KongDepartment of Physics, Ningbo University, Ningbo, Zhejiang, ChinaExact periodic and localized solutions of a nonlocal Mel′nikov equation are derived by the Hirota bilinear method. Many conventional nonlocal operators involve integration over a spatial or temporal domain. However, the present class of nonlocal equations depends on properties at selected far field points which result in a potential satisfying parity time symmetry. The present system of nonlocal partial differential equations consists of two dependent variables in two spatial dimensions and time, where the dependent variables physically represent a wave packet and an auxiliary scalar field. The periodic solutions may take the forms of breathers (pulsating modes) and line solitons. The localized solutions can include propagating lumps and rogue waves. These nonsingular solutions are obtained by appropriate choice of parameters in the Hirota expansion. Doubly periodic solutions are also computed with elliptic and theta functions. In sharp contrast with the local Mel′nikov equation, the auxiliary scalar field in the present set of solutions can attain complex values. Through a coordinate transformation, the governing equation can reduce to the Schrödinger–Boussinesq system.http://dx.doi.org/10.1155/2020/2642654
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Wei Liu Zhenyun Qin Kwok Wing Chow Senyue Lou
spellingShingle	Wei Liu Zhenyun Qin Kwok Wing Chow Senyue Lou Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation Complexity
author_facet	Wei Liu Zhenyun Qin Kwok Wing Chow Senyue Lou
author_sort	Wei Liu
title	Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation
title_short	Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation
title_full	Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation
title_fullStr	Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation
title_full_unstemmed	Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation
title_sort	families of rational and semirational solutions of the partial reverse space-time nonlocal mel′nikov equation
publisher	Hindawi-Wiley
series	Complexity
issn	1076-2787 1099-0526
publishDate	2020-01-01
description	Exact periodic and localized solutions of a nonlocal Mel′nikov equation are derived by the Hirota bilinear method. Many conventional nonlocal operators involve integration over a spatial or temporal domain. However, the present class of nonlocal equations depends on properties at selected far field points which result in a potential satisfying parity time symmetry. The present system of nonlocal partial differential equations consists of two dependent variables in two spatial dimensions and time, where the dependent variables physically represent a wave packet and an auxiliary scalar field. The periodic solutions may take the forms of breathers (pulsating modes) and line solitons. The localized solutions can include propagating lumps and rogue waves. These nonsingular solutions are obtained by appropriate choice of parameters in the Hirota expansion. Doubly periodic solutions are also computed with elliptic and theta functions. In sharp contrast with the local Mel′nikov equation, the auxiliary scalar field in the present set of solutions can attain complex values. Through a coordinate transformation, the governing equation can reduce to the Schrödinger–Boussinesq system.
url	http://dx.doi.org/10.1155/2020/2642654
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Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation

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