Eckhaus equation
Appearance
In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equation within the nonlinear Schrödinger class:[1]
The equation was independently introduced by Wiktor Eckhaus and by Anjan Kundu to model the propagation of waves in dispersive media.[2][3]
Linearization
![](http://upload.wikimedia.org/wikipedia/commons/4/4f/Eckhaus_equation_wave_packet.gif)
![](http://upload.wikimedia.org/wikipedia/commons/1/1f/Schr%C3%B6dinger_equation_wave_packet.gif)
The Eckhaus equation can be linearized to the linear Schrödinger equation:[4]
through the non-linear transformation:[5]
The inverse transformation is:
This linearization also implies that the Eckhaus equation is integrable.
Notes
References
- S2CID 59441129
- S2CID 250876392
- Eckhaus, W. (1985), The long-time behaviour for perturbed wave-equations and related problems, Department of Mathematics, University of Utrecht, Preprint no. 404.
Published in part in: Eckhaus, W. (1986), "The long-time behaviour for perturbed wave-equations and related problems", in Kröner, E.; Kirchgässner, K. (eds.), Trends in applications of pure mathematics to mechanics, Lecture Notes in Physics, vol. 249, Berlin: Springer, pp. 168–194,ISBN 978-3-540-16467-8 - Kundu, A. (1984), "Landau–Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations", Journal of Mathematical Physics, 25 (12): 3433–3438, doi:10.1063/1.526113
- Taghizadeh, N.; Mirzazadeh, M.; Tascan, F. (2012), "The first-integral method applied to the Eckhaus equation", Applied Mathematics Letters, 25 (5): 798–802,
- Zwillinger, D. (1998), Handbook of differential equations (3rd ed.), Academic Press, ISBN 978-0-12-784396-4