### 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 language | English |
---|---|

Pages (from-to) | 44-66 |

Number of pages | 23 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 176 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2003 Feb 15 |

Externally published | Yes |

### Fingerprint

### 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

*Physica D: Nonlinear Phenomena*,

*176*(1-2), 44-66. https://doi.org/10.1016/S0167-2789(02)00708-X

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

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 176, no. 1-2, pp. 44-66. https://doi.org/10.1016/S0167-2789(02)00708-X

}

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 -