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In [[mathematical physics]], the '''Eckhaus equation''' |
In [[mathematical physics]], the '''Eckhaus equation''', a nonlinear [[partial differential equations]] within the [[nonlinear Schrödinger equation|nonlinear Schrödinger]] class:<ref>{{harvtxt|Zwillinger|1998|pp=177 & 390}}</ref> |
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:<math>i \psi_t + \psi_{xx} +2 \left( |\psi|^2 \right)_x\, \psi + |\psi|^4\, \psi = 0.</math> |
:<math>i \psi_t + \psi_{xx} +2 \left( |\psi|^2 \right)_x\, \psi + |\psi|^4\, \psi = 0.</math> |
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was introduced by [[Wiktor Eckhaus]] to model the propagation of [[wave]]s in [[dispersion (optics)|dispersive media]].<ref>{{harvtxt|Eckhaus|1985}}</ref>. A generalization of this equation with additiobal cubic nonlinear term was proposed earlier by Anjan Kundu through gauge equivalence from the [[nonlinear Schrödinger equation]]<ref>{{harvtxt|Kundu|1984}}</ref>, which is now known as the '''Kundu-Eckhaus equation'''. |
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==Linearization== |
==Linearization== |
Revision as of 10:30, 25 April 2016
In mathematical physics, the Eckhaus equation, a nonlinear partial differential equations within the nonlinear Schrödinger class:[1]
was introduced by Wiktor Eckhaus to model the propagation of waves in dispersive media.[2]. A generalization of this equation with additiobal cubic nonlinear term was proposed earlier by Anjan Kundu through gauge equivalence from the nonlinear Schrödinger equation[3], which is now known as the Kundu-Eckhaus equation.
Linearization
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
- Ablowitz, M.J.; Ahrens, C.D.; De Lillo, S. (2005), "On a "quasi" integrable discrete Eckhaus equation", Journal of Nonlinear Mathematical Physics, 12 (Supplement 1): 1–12, Bibcode:2005JNMP...12S...1A, doi:10.2991/jnmp.2005.12.s1.1
- Calogero, F.; De Lillo, S. (1987), "The Eckhaus PDE iψt + ψxx+ 2(|ψ|2)x ψ + |ψ|4 ψ = 0", Inverse Problems, 3 (4): 633–682, Bibcode:1987InvPr...3..633C, doi:10.1088/0266-5611/3/4/012
- 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, doi:10.1007/BFb0016391, 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: 3433–3438, Bibcode:1984JMP....25.3433K, 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, doi:10.1016/j.aml.2011.10.021
- Zwillinger, D. (1998), Handbook of differential equations (3rd ed.), Academic Press, ISBN 978 0 12 784396 4