Abstract
It is shown that the complete exceptionality condition for discontinuity waves associated with a second-order non-linear hyperbolic equation of the form
leads to a Monge-Ampère-type equation in 3+1 dimensions. Application of a novel reciprocal transformation shows that an important subclass may be reduced to linear canonical form. Specialization to 1+1 dimensions yields linearization of a Boillat-type equation satisfying the complete exceptionality criterion. In this last case the transformation allowing the linearization coincide with the one introduced by Hoskins and Bretherton in the theory of atmospheric frontogenesis and so-called geostrophic transformation. Finally, always in 1+1 dimensions, we show that the Monge-Ampère equation is also strictly exceptional, i.e. the only possible shocks are characteristic.
Sommario
Si dimostra che la condizione di completa eccezionalita' per le onde di discontinuita' associate con una equazione non-lineare iperbolica della forma:
e' soddisfatta se l'equazione e' di tipo Monge-Ampére in 3+1 dimensioni. Inoltra, esiste una transformazione reciproca che riduce una sottoclasse di tali equazioni a forma canonica lineare. La particolarizzazione di tale transformazione al caso di 1+1 dimensioni linearizza l'equazione non-lineare del secondo ordine, che gode della proprieta' di essere completamente eccezionale, ottenuta da Boillat. Tale trasformazione coincide con quella detta geostrofica e introdotta da Hoskins e Bretherton nella teoria della frontogenesi atmosferica. Infine, sempre nel caso a 1+1 dimensioni, si dimostra che l'equazione di Monge-Ampére presenta anche il carattere di stretta eccezionalita' cioe' i soli urti possibili si propagano con velocita' caratteristiche.
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Donato, A., Ramgulam, U. & Rogers, C. The (3+1)-dimensional Monge-Ampère equation in discontinuity wave theory: Application of a reciprocal transformation. Meccanica 27, 257–262 (1992). https://doi.org/10.1007/BF00424364
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DOI: https://doi.org/10.1007/BF00424364