In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

within the nonlinear Schrödinger class: :

i \psi_t + \psi_ +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.

Linearization

The Eckhaus equation can be linearized to the linear Schrödinger equation: :

i \varphi_t + \varphi_ =0,

through the non-linear transformation: :

\varphi(x,t) = \psi(x,t)\, \exp\left( \int_^x , \psi(x^\prime,t), ^2\; \textx^\prime \right).

The inverse transformation is: :

\psi(x,t) = \frac.

This linearization also implies that the Eckhaus equation is integrable.

Notes

References

* * *.
Published in part in: * * * {{citation , title=Handbook of differential equations , first=D. , last=Zwillinger , edition=3rd , publisher=Academic Press , year=1998 , isbn=978-0-12-784396-4 Nonlinear partial differential equations Schrödinger equation