Painlevé analysis and exact solutions of nonintegrable systems.

*(English)* Zbl 1100.35099
Mladenov, Ivaïlo (ed.) et al., Proceedings of the 7th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 2–10, 2005. Sofia: Bulgarian Academy of Sciences (ISBN 954-8495-30-9/pbk). 280-291 (2006).

Summary: We consider the cubic complex Ginzburg-Landau equation. Applying Hone’s method, based on the use of the Laurent series solutions and the residue theorem, we prove that this equation has no elliptic standing wave solutions. This result supplements Hone’s, result, that this equation has no elliptic travelling wave solutions. It is shown that Hone’s method can be applied to a system of polynomial differential equations more effectively than to an equivalent differential equation.

##### MSC:

35Q55 | NLS-like (nonlinear Schrödinger) equations |

37K20 | Relations of infinite-dimensional systems with algebraic geometry, etc. |

30B50 | Dirichlet series, other series expansions, exponential series (one complex variable) |