Summary
The general similarity solution of the Boussinesq equation is expressed in terms of the first Painlevé transcendents or of the Weierstrass elliptic functions. A new more general Bäcklund transformation (BT) with three parameters has been found. The BT applied to similarity solutions cannot map first Painlevé transcendents into each other, but can map Weierstrass functions into each other. In this case the BT is the addition formula for the Weierstrass ζ-functions. As a by-product of the investigation on the BT the general solution for the stationary linear Schrödinger equation with potential a WeierstrassP-function is obtained.
Riassunto
Si esprime la soluzione generale di similarità dell'equazione di Boussinesq tramite i trascendenti di Painlevé di primo tipo oppure tramite le funzioni ellittiche di Weierstrass. Si trova una nuova trasformata di Bäcklund (BT), più generale, scritta in funzione di tre parametri. La BT, applicata alle soluzioni di similarità, non trasforma i trascendenti di Painlevé di primo tipo in se stessi, però è in grado di trasformare funzioni di Weierstrass in se stesse. In questo caso la BT diventa la formula di addizione per le fuzioni ζ di Weierstrass. Infine, come prodotto secondario nello studio della BT, si ottiene la soluzione generale dell'equazione lineare e stazionaria di Schrödinger con un potenziale che abbia la forma di una funzioneP di Weierstrass.
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Boiti, M., Pempinelli, F. Similarity solutions and Bäcklund transformations of the Boussinesq equation. Nuov Cim B 56, 148–156 (1980). https://doi.org/10.1007/BF02738364
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DOI: https://doi.org/10.1007/BF02738364