Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation

Kenichi Maruno, Adrian Ankiewicz, Nail Akhmediev

Research output: Contribution to journalArticle

29 Citations (Scopus)

Abstract

Using Painlevé analysis, the Hirota multi-linear method and a direct ansatz technique, we study analytic solutions of the (1+1)-dimensional complex cubic and quintic Swift-Hohenberg equations. We consider both standard and generalized versions of these equations. We have found that a number of exact solutions exist to each of these equations, provided that the coefficients are constrained by certain relations. The set of solutions include particular types of solitary wave solutions, hole (dark soliton) solutions and periodic solutions in terms of elliptic Jacobi functions and the Weierstrass ℘ function. Although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic Swift-Hohenberg equations, those presented here are the first examples of exact analytic solutions found thus far.

Original languageEnglish
Pages (from-to)44-66
Number of pages23
JournalPhysica D: Nonlinear Phenomena
Volume176
Issue number1-2
DOIs
Publication statusPublished - 2003 Feb 15
Externally publishedYes

Fingerprint

Swift-Hohenberg Equation
Quintic
Soliton Solution
Solitons
Analytic Solution
solitary waves
Weierstrass Function
Jacobi Elliptic Function
Solitary Wave Solution
Particular Solution
Periodic Solution
Weierstrass functions
Exact Solution
Subset
Coefficient
set theory
coefficients
Standards

Keywords

  • Complex Swift-Hohenberg equation
  • Direct ansatz method
  • Hirota multi-linear method
  • Singularity analysis
  • Solitons

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation. / Maruno, Kenichi; Ankiewicz, Adrian; Akhmediev, Nail.

In: Physica D: Nonlinear Phenomena, Vol. 176, No. 1-2, 15.02.2003, p. 44-66.

Research output: Contribution to journalArticle

Maruno, Kenichi ; Ankiewicz, Adrian ; Akhmediev, Nail. / Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation. In: Physica D: Nonlinear Phenomena. 2003 ; Vol. 176, No. 1-2. pp. 44-66.
@article{d012c180dbf64ee5968285ab149ac6ba,
title = "Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation",
abstract = "Using Painlev{\'e} analysis, the Hirota multi-linear method and a direct ansatz technique, we study analytic solutions of the (1+1)-dimensional complex cubic and quintic Swift-Hohenberg equations. We consider both standard and generalized versions of these equations. We have found that a number of exact solutions exist to each of these equations, provided that the coefficients are constrained by certain relations. The set of solutions include particular types of solitary wave solutions, hole (dark soliton) solutions and periodic solutions in terms of elliptic Jacobi functions and the Weierstrass ℘ function. Although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic Swift-Hohenberg equations, those presented here are the first examples of exact analytic solutions found thus far.",
keywords = "Complex Swift-Hohenberg equation, Direct ansatz method, Hirota multi-linear method, Singularity analysis, Solitons",
author = "Kenichi Maruno and Adrian Ankiewicz and Nail Akhmediev",
year = "2003",
month = "2",
day = "15",
doi = "10.1016/S0167-2789(02)00708-X",
language = "English",
volume = "176",
pages = "44--66",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",
number = "1-2",

}

TY - JOUR

T1 - Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation

AU - Maruno, Kenichi

AU - Ankiewicz, Adrian

AU - Akhmediev, Nail

PY - 2003/2/15

Y1 - 2003/2/15

N2 - Using Painlevé analysis, the Hirota multi-linear method and a direct ansatz technique, we study analytic solutions of the (1+1)-dimensional complex cubic and quintic Swift-Hohenberg equations. We consider both standard and generalized versions of these equations. We have found that a number of exact solutions exist to each of these equations, provided that the coefficients are constrained by certain relations. The set of solutions include particular types of solitary wave solutions, hole (dark soliton) solutions and periodic solutions in terms of elliptic Jacobi functions and the Weierstrass ℘ function. Although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic Swift-Hohenberg equations, those presented here are the first examples of exact analytic solutions found thus far.

AB - Using Painlevé analysis, the Hirota multi-linear method and a direct ansatz technique, we study analytic solutions of the (1+1)-dimensional complex cubic and quintic Swift-Hohenberg equations. We consider both standard and generalized versions of these equations. We have found that a number of exact solutions exist to each of these equations, provided that the coefficients are constrained by certain relations. The set of solutions include particular types of solitary wave solutions, hole (dark soliton) solutions and periodic solutions in terms of elliptic Jacobi functions and the Weierstrass ℘ function. Although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic Swift-Hohenberg equations, those presented here are the first examples of exact analytic solutions found thus far.

KW - Complex Swift-Hohenberg equation

KW - Direct ansatz method

KW - Hirota multi-linear method

KW - Singularity analysis

KW - Solitons

UR - http://www.scopus.com/inward/record.url?scp=0037441006&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037441006&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(02)00708-X

DO - 10.1016/S0167-2789(02)00708-X

M3 - Article

VL - 176

SP - 44

EP - 66

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -