Two modifications of Ito's equation

Integration via Modification: A Method of Reduction of Order for Systems of Ordinary Differential Equations

“…We note that the integrated modified equation results from the substitution of u = F[U] into (41), or equivalently into (42). Thus we see that, provided that L N −1 [u] = 0, and provided that the relations (36) are consistent -for which a sufficient condition, for example, is that ε 0 = 0 -we are always able to effect a reduction of the order of our system (32) by N . This is true whatever the form of L[u].…”

“…and substitution into (40) then gives the integrated version of equation (32). We note that the integrated modified equation results from the substitution of u = F[U] into (41), or equivalently into (42).…”

“…We note that our calculation of the above integrated system is valid provided that the relations ε 0 = α 2 0 , ε 1 = 2α 0 α 1 are consistent. The only case in which these relations are not consistent is ε 0 = 0 = ε 1 , which corresponds in the PDE case to Ito's equation [31], and for which the difficulties of obtaining a Miura map are well known [30,32]; see also Section 4.3. However, we note that the above formulae still give an integrated version of the system (49), (50), even in this troublesome case.…”

“…It is the factorization of this Hamiltonian operator that allows the construction of modifications of Ito's equation [32]. This factorization reads…”

Systematic method of generating new integrable systems via inverse Miura maps

“…Thus, we may call (3.1) or (3.2) the Zakharov-Ito system. The Lax representation for (3.2) is given by the pair of linear equations for a scalar function ψ [13,64], [65] for the ζ = 1 case). Substituting this expression for r into (3.2a) and combining it with (3.4b), we obtain the modified system for (q, v),…”