Eckhaus equation
# Eckhaus equation

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

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

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

## Linearization

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

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

through the non-linear transformation:

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

The inverse transformation is:

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

This linearization also implies that the Eckhaus equation is integrable.

