Geodesic
A quest for the integrable equation in $3+1$ dimensions
Teoretičeskaâ i matematičeskaâ fizika, Tome 122 (2000) no. 2, pp. 305-309

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The $L$ and $T$ operators of the Korteweg–de Vries equation are modified to seek a $(3+1)$-dimensional integrable equation. However, the Lax equation in this case is eventually reduced to a $(2+1)$-dimensional equation. We also propose other modified equations and their Lax pairs. A similar attempt is made to derive a higher-dimensional Harry Dym (HD) equation. As a result, a new $(2+1)$-dimensional HD equation is presented. 
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     author = {Yu. Song-Ju and K. Toda and T. Fukuyama},
     title = {A quest for the integrable equation in $3+1$ dimensions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {305--309},
     publisher = {mathdoc},
     volume = {122},
     number = {2},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2000_122_2_a12/}
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Yu. Song-Ju; K. Toda; T. Fukuyama. A quest for the integrable equation in $3+1$ dimensions. Teoretičeskaâ i matematičeskaâ fizika, Tome 122 (2000) no. 2, pp. 305-309. http://geodesic.mathdoc.fr/item/TMF_2000_122_2_a12/