Research ArticleOpen AccessGoogle Scholar indexed
Traveling Wave Solutions of the Quintic Complex One-Dimensional Ginzburg-Landau Equation
Department of Physics, University of Osnabrück, Osnabrück, Germany
Research Unit of Mathematical Sciences, University of Oulu, Finland and Moscow Centre of Fundamental and Applied Mathematics—MSU, Moscow, Russia
- 1 Department of Physics, University of Osnabrück, Osnabrück, Germany
- 2 Research Unit of Mathematical Sciences, University of Oulu, Finland and Moscow Centre of Fundamental and Applied Mathematics—MSU, Moscow, Russia
Copy link · social · email
Abstract
A subset of traveling wave solutions of the quintic complex Ginzburg-Landau equation (QCGLE) is presented in compact form. The approach consists of the following parts: 1) Reduction of the QCGLE to a system of two ordinary differential equations (ODEs) by a traveling wave ansatz; 2) Solution of the system for two (ad hoc) cases relating phase and amplitude; 3) Presentation of the solution for both cases in compact form; 4) Presentation of constraints for bounded and for singular positive solutions by analysing the analytical properties of the solution by means of a phase diagram approach. The results are exemplified numerically.
KeywordsGinzburg-Landau EquationWeierstrass’ Elliptic FunctionPhase Diagram
- Aranson, I. and Kramer, L. (2001) The World of the Complex Ginzburg-Landau Equation. Reviews of Modern Physics, 74, 99-143. (Preprint cond-mat/0106115) https://doi.org/10.1103/RevModPhys.74.99
- van Saarloos, W. and Hohenberg, P.C. (1992) Fronts, Pulses, Sources and Sinks in the Generalised Complex Ginzburg-Landau Equation. Physica D: Nonlinear Phenomena, 56, 303-367. https://doi.org/10.1016/0167-2789(92)90175-M
- Osman, M.C., Ghanbari, B. and Machado, J.A.T. (2019) New Complex Waves in Nonlinear Optics Based on the Complex Ginzburg-Landau Equation with Kerr Low Nonlinearity. The European Physical Journal Plus, 134, Article No. 20. https://doi.org/10.1140/epjp/i2019-12442-4
- (a) Musette, M. and Conte, R. (2003) Analytic Solitary Waves of Nonintegrable Systems. Physica D: Nonlinear Phenomena, 181, 70-79. (Preprint nlin.PS/0302051) https://doi.org/10.1016/S0167-2789(03)00069-1 (b) Conte, R. and Musette, M. (2009) Elliptic General Analytic Solutions. Studies in Applied Mathematics, 123, 63-81. https://doi.org/10.1111/j.1467-9590.2009.00447.x
- Vernov Yu, S. (2007) Elliptic Solutions of the Quintic Complex One-Dimensional Ginzburg-Landau Equation. Journal of Physics A: Mathematical and Theoretical, 40, 9833-9844. https://doi.org/10.1088/1751-8113/40/32/009
- Conte, R. and Ng, T.W. (2012) Meromorphic Traveling Wave Solutions of the Complex Cubic-Quintic Ginzburg-Landau Equation. Acta Applicandae Mathematicae, 127, 153-166. https://doi.org/10.1007/s10440-012-9734-y
- Conte, R. and Ng, T.W. (2012) Detection and Construction of an Elliptic Solution of the Complex Cubic-Quintic Ginzburg-Landau Equation. Theoretical and Mathematical Physics, 172, 1073-1084. https://doi.org/10.1007/s11232-012-0096-4
- Akhmediev, N.N. and Afanasjev, V.V. (1996) Singularities and Special Soliton Solutions of the Cubic-Quintic Complex Ginzburg-Landau Equation. Physical Review E, 53, 1190-1201. https://doi.org/10.1103/PhysRevE.53.1190
- Marcq, Ph., Chate, H. and Conte, R. (1994) Exact Solutions of the One Dimensional Quintic Complex Ginzburg-Landau Equation. Physica D: Nonlinear Phenomena, 73, 305-317. (Preprint patt-sol/931004) https://doi.org/10.1016/0167-2789(94)90102-3
- van Saarloos, W. and Hohenberg, P.C. (1990) Pulses and Fronts in the Complex Ginzburg-Landau Equation Near a Subcritical Bifurcation. Physical Review Letters, 64, 749-752. https://doi.org/10.1103/PhysRevLett.64.749
- Doelman, A. (1996) Traveling Waves in the Complex Ginzburg-Landau Equation. Physica D: Nonlinear Phenomena, 97, 398-428. https://doi.org/10.1016/0167-2789(95)00303-7