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# On higher order isolas of unstable Stokes waves
This project is companion  to the paper [On higher order isolas of unstable Stokes waves](https://people.sissa.it/~amaspero/research.html) by M. Berti, L. Corsi, A. Maspero and P. Ventura.
We collect Mathematica codes to compute the asymptotic expansion, as the depth $h$ goes to infinity,  of the coefficients $\beta^{(p)}_1(h)$ for $p=2, 3, 4$. 
The code has been developed in Mathematica   by  P. Ventura.  Feel free to contact us at `amaspero@sissa.it` and `paolo.ventura@unimi.it`!
## NB: Usage
To prevent multiple labeling, it is essential to restart the Mathematica kernel after evaluating each notebook. 
### Notebooks
1)  `conto_p=2.nb` is the Mathematica notebook to compute the asymptotic expansion of $\beta_1^{(2)}(h)$ as the depth $h$ goes to infinity 
2)  `conto_p=3.nb` is the Mathematica notebook to compute the asymptotic expansion of $\beta_1^{(3)}(h)$ as the depth $h$ goes to infinity 
3)  `conto_p=4.nb` is the Mathematica notebook to compute the asymptotic expansion of $\beta_1^{(4)}(h)$ as the depth $h$ goes to infinity 
### Libraries
- `CoeffsLin.m` contains the coefficients of the functions ${p_\epsilon(x)}$ and ${a_\epsilon(x)}$ obtained in the paper [[2]](#2)-  The library is used in the notebooks `beta1.nb`.
## References
<a id="1">[1]</a> 
M. Berti, L. Corsi, A. Maspero and P. Ventura. 
_Infinitely many isolas of modulational instability of Stokes waves_, arXiv 2024
<a id="1">[2]</a> 
M. Berti, L. Corsi, A. Maspero and P. Ventura. 
_On higher order isolas of unstable Stokes waves_, arXiv 2025
<a id="2">[3]</a> 
M. Berti, A. Maspero and P. Ventura. 
_Stokes Waves at the Critical Depth are Modulationally Unstable_, CMP,   405(56), 2024