Novel homoclinic and heteroclinic solutions for the 2D complex cubic Ginzburg–Landau equation

Jian HUANG; Mingming LENG; Zhengde DAI

doi:10.1016/j.physleta.2009.10.069

Novel homoclinic and heteroclinic solutions for the 2D complex cubic Ginzburg–Landau equation

Jian HUANG, Mingming LENG, Zhengde DAI

Department of Computing and Decision Sciences

Research output: Journal Publications › Journal Article (refereed) › peer-review

1 Citation (Scopus)

Abstract

Homoclinic and heteroclinic solutions are two important concepts that are used to investigate the complex properties of nonlinear evolutionary equations. In this Letter, we perform hyperbolic and linear stability analysis, and prove the existence of homoclinic and heteroclinic solutions for two-dimensional cubic Ginzburg–Landau equation with periodic boundary condition and even constraint. Then, using the Hirota's bilinear transformation, we find the closed-form homoclinic and heteroclinic solutions. Moreover, we find that the homoclinic tubes and two families of heteroclinic solutions are asymptotic to a periodic cycle in one dimension.

Original language	English
Pages (from-to)	258-263
Number of pages	6
Journal	Physics Letters A
Volume	374
Issue number	2
DOIs	https://doi.org/10.1016/j.physleta.2009.10.069
Publication status	Published - 28 Dec 2009

Keywords

2D cubic Ginzburg–Landau equation; Homoclinic orbits; Heteroclinic orbits; Hyperbolic property; Linearized stability; Hirota's bilinear transformation

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10.1016/j.physleta.2009.10.069

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title = "Novel homoclinic and heteroclinic solutions for the 2D complex cubic Ginzburg–Landau equation",

abstract = "Homoclinic and heteroclinic solutions are two important concepts that are used to investigate the complex properties of nonlinear evolutionary equations. In this Letter, we perform hyperbolic and linear stability analysis, and prove the existence of homoclinic and heteroclinic solutions for two-dimensional cubic Ginzburg–Landau equation with periodic boundary condition and even constraint. Then, using the Hirota's bilinear transformation, we find the closed-form homoclinic and heteroclinic solutions. Moreover, we find that the homoclinic tubes and two families of heteroclinic solutions are asymptotic to a periodic cycle in one dimension.",

keywords = "2D cubic Ginzburg–Landau equation; Homoclinic orbits; Heteroclinic orbits; Hyperbolic property; Linearized stability; Hirota's bilinear transformation",

author = "Jian HUANG and Mingming LENG and Zhengde DAI",

year = "2009",

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T1 - Novel homoclinic and heteroclinic solutions for the 2D complex cubic Ginzburg–Landau equation

AU - HUANG, Jian

AU - LENG, Mingming

AU - DAI, Zhengde

PY - 2009/12/28

Y1 - 2009/12/28

N2 - Homoclinic and heteroclinic solutions are two important concepts that are used to investigate the complex properties of nonlinear evolutionary equations. In this Letter, we perform hyperbolic and linear stability analysis, and prove the existence of homoclinic and heteroclinic solutions for two-dimensional cubic Ginzburg–Landau equation with periodic boundary condition and even constraint. Then, using the Hirota's bilinear transformation, we find the closed-form homoclinic and heteroclinic solutions. Moreover, we find that the homoclinic tubes and two families of heteroclinic solutions are asymptotic to a periodic cycle in one dimension.

AB - Homoclinic and heteroclinic solutions are two important concepts that are used to investigate the complex properties of nonlinear evolutionary equations. In this Letter, we perform hyperbolic and linear stability analysis, and prove the existence of homoclinic and heteroclinic solutions for two-dimensional cubic Ginzburg–Landau equation with periodic boundary condition and even constraint. Then, using the Hirota's bilinear transformation, we find the closed-form homoclinic and heteroclinic solutions. Moreover, we find that the homoclinic tubes and two families of heteroclinic solutions are asymptotic to a periodic cycle in one dimension.

KW - 2D cubic Ginzburg–Landau equation; Homoclinic orbits; Heteroclinic orbits; Hyperbolic property; Linearized stability; Hirota's bilinear transformation

UR - http://commons.ln.edu.hk/sw_master/1615

UR - http://www.scopus.com/inward/record.url?scp=71849115354&partnerID=8YFLogxK

U2 - 10.1016/j.physleta.2009.10.069

DO - 10.1016/j.physleta.2009.10.069

M3 - Journal Article (refereed)

SN - 0375-9601

VL - 374

SP - 258

EP - 263

JO - Physics Letters A

JF - Physics Letters A

IS - 2

ER -

Novel homoclinic and heteroclinic solutions for the 2D complex cubic Ginzburg–Landau equation

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