The lump solutions and the Bäcklund transformation for the three‐dimensional three‐wave resonant interaction

A Backlund transformation is found for the three‐dimensional three‐wave resonant interaction, and from it, N‐lump exact solutions may be constructed. The one‐lump solution is analyzed in detail, and it is shown that it describes such effects as pulse decay, upconversion, and explosive instabilities, all in three dimensions.