Summary
In this paper we find the general similarity solution for the Korteweg-de Vries equation in terms of the second transcendents of Painlevé. It is shown that the Bäcklund transformation for the similarity solutions reduces to an algebraic transformation on the derivatives of the starting solution. The Bäcklund transformation is expressed in the language of the second Painlevé transcendents, thus obtaining new interesting properties for these special functions. Some explicit classes of solutions of the Korteweg-de Vries equation are obtained and described.
Riassunto
In questo lavoro si ricava la soluzione generale di similarità per l'equazione di Kortewegde Vries, esprimendola attraverso i secondi trascendenti di Painlevé. Si dimostra che la trasformazione di Bäcklund per le soluzioni di similarità si riduce ad una trasformazione algebrica sulle derivate della soluzione iniziale. La trasformata di Bäcklund è espressa nel linguaggio dei secondi trascendenti di Painlevé e così si ottengono nuove proprietà interessanti per queste funzioni speciali. Sono inoltre ricavate e descritte alcune classi esplicite di soluzioni per l'equazione di Korteweg-de Vries.
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Boiti, M., Pempinelli, F. Similarity solutions of the Korteweg-de Vries equation. Nuov Cim B 51, 70–78 (1979). https://doi.org/10.1007/BF02743697
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DOI: https://doi.org/10.1007/BF02743697