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A quest for the integrable equation in \(3+1\) dimensions. (English. Russian original) Zbl 0981.37025
Theor. Math. Phys. 122, No. 2, 256-259 (2000); translation from Teor. Mat. Fiz. 122, No. 2, 305-309 (2000).
Summary: 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.
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems