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

Jian HUANG, Mingming LENG, Zhengde DAI

Research output: Journal PublicationsJournal Article (refereed)Researchpeer-review

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 languageEnglish
Pages (from-to)258-263
Number of pages6
JournalPhysics Letters A
Volume374
Issue number2
DOIs
Publication statusPublished - 28 Dec 2009

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cubic equations
nonlinear equations
boundary conditions
tubes
cycles

Keywords

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

Cite this

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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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language = "English",
volume = "374",
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journal = "Physics Letters, Section A: General, Atomic and Solid State Physics",
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}

Novel homoclinic and heteroclinic solutions for the 2D complex cubic Ginzburg–Landau equation. / HUANG, Jian; LENG, Mingming; DAI, Zhengde.

In: Physics Letters A, Vol. 374, No. 2, 28.12.2009, p. 258-263.

Research output: Journal PublicationsJournal Article (refereed)Researchpeer-review

TY - JOUR

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

U2 - 10.1016/j.physleta.2009.10.069

DO - 10.1016/j.physleta.2009.10.069

M3 - Journal Article (refereed)

VL - 374

SP - 258

EP - 263

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

SN - 0375-9601

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