Eckhaus equation

In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equation within the nonlinear Schrödinger class:^[1]

${\displaystyle i\psi _{t}+\psi _{xx}+2\left(|\psi |^{2}\right)_{x}\,\psi +|\psi |^{4}\,\psi =0.}$

The equation was independently introduced by Wiktor Eckhaus and by Anjan Kundu to model the propagation of waves in dispersive media.^[2][3]

Linearization

The Eckhaus equation can be linearized to the linear Schrödinger equation:^[4]

${\displaystyle i\varphi _{t}+\varphi _{xx}=0,}$

through the non-linear transformation:^[5]

${\displaystyle \varphi (x,t)=\psi (x,t)\,\exp \left(\int _{-\infty }^{x}|\psi (x^{\prime },t)|^{2}\;{\text{d}}x^{\prime }\right).}$

The inverse transformation is:

${\displaystyle \psi (x,t)={\frac {\varphi (x,t)}{\displaystyle \left(1+2\,\int _{-\infty }^{x}|\varphi (x^{\prime },t)|^{2}\;{\text{d}}x^{\prime }\right)^{1/2}}}.}$

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, S2CID 59441129
• Calogero, F.; De Lillo, S. (1987), "The Eckhaus PDE iψ_t + ψ_xx+ 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, 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, 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 (12): 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

This page was last edited on 29 January 2023, at 03:13