mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

, 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

wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...

s 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