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 | |
| Publication status | Published - 28 Dec 2009 |
Funding
The authors are grateful to an anonymous referee for his or her insightful comments that helped improve the Letter. This work is supported by NSFC (70901036) and the Croucher Foundation (RSD163/0809/S).
Keywords
- 2D cubic Ginzburg–Landau equation
- Homoclinic orbits
- Heteroclinic orbits
- Hyperbolic property
- Linearized stability
- Hirota’s bilinear transformation
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