Abstract
The Kadomtsev–Petviashvili equation is known to be the leading term of a semi-infinite hierarchy of integrable equations with evolutions given by times with positive numbers. Here, we introduce new hierarchy directed to negative numbers of times. The derivation of such systems, as well as the corresponding hierarchy, is based on the commutator identities. This approach enables introduction of linear differential equations that admit lifts up to nonlinear integrable ones by means of the special dressing procedure. Thus, one can construct not only nonlinear equations, but corresponding Lax pairs as well. The Lax operator of this evolution coincides with the Lax operator of the “positive” hierarchy. We also derive (1 + 1)-dimensional reductions of equations of this hierarchy.
Highlights
The main examples of (2 + 1) dimensional integrable hierarchies appear due to Zakharov–Shabat systems [1], or the approach of Miwa–Jimbo–Date [2], as semi-infinite sets of equations with a common Lax operator
By means of the Kadomtsev–Petviashvili (KP) equation [3], we derive new kinds of integrable hierarchies that can be associated to negative numbers of times
Davey–Stewartson (DS) hierarchy [5] was considered. Construction of such hierarchies gives an essential extension of the set of integrable equations because the approach of [1,2]
Summary
The main examples of (2 + 1) dimensional integrable hierarchies appear due to Zakharov–Shabat systems [1], or the approach of Miwa–Jimbo–Date [2], as semi-infinite sets of equations with a common Lax operator. These sequences start with the lowest (first) equations, and the numbers of times grow together with the order of the second Lax operators. By means of the Kadomtsev–Petviashvili (KP) equation [3], we derive new kinds of integrable hierarchies that can be associated to negative numbers of times This approach was suggested in [4], where example of the. Construction of such hierarchies gives an essential extension of the set of integrable equations because the approach of [1,2]
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