# Solution of the complex Ginzburg - Landau equation

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

-

## 1 Answer

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 $\Phi \left( x,t \right) =R{{\rm e}^{i \left( \sqrt {1+{R}^{2}}x+ \left( a \left( 1+{R}^{2} \right) -b{R}^{2} \right) t \right) }}$

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 8, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9160796403884888, "perplexity_flag": "head"}