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Can someone show that it's possible to find a solution of the kind:

$$\Phi(x,t)=R(x,t)\exp[i\Psi(x,t)]$$

of the complex Ginzburg - Landau equation:

$$\frac{\partial{\Phi}}{\partial{t}}=(1+ia)\frac{\partial^2{\Phi}}{\partial{x}^2}+\Phi-(ib-1)|\Phi|^2\Phi$$

assuming that $R(x,t)$ and $\Psi(x,t)$

are defined as real - valued functions?

Thanks

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up vote 1 down vote accepted

Any complex number $z$ can be written as $z = r e^{i\theta}$ where $r = |z| > 0$ and $\theta$ is real, so any solution $\Phi(x,t)$ can certainly be written that way.

Some particular solutions in this case are $\Phi \left( x,t \right) =R{{\rm e}^{i \left( \sqrt {1+{R}^{2}}x+ \left( a \left( 1+{R}^{2} \right) -b{R}^{2} \right) t \right) }} $

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