Zakharov-Ito equation and Generalized Heisenberg ferromagnet-type equation: equivalence and related geometric curve flows

TL;DR

This paper demonstrates the gauge and geometric equivalence between the Zakharov-Ito equation and a generalized Heisenberg ferromagnet-type equation, constructing related geometric curve flows and deriving explicit soliton solutions.

Contribution

It establishes the gauge and geometric equivalence between ZIE and GHFE and constructs integrable space curve flows with explicit soliton solutions.

Findings

ZIE and GHFE are gauge and geometrically equivalent.

Constructed integrable motions of space curves induced by ZIE.

Derived the 1-soliton solution of GHFE from ZIE seed solution.

Abstract

These results continue our studies of integrable generalized Heisenberg ferromagnet-type equations (GHFE) and their equivalent counterparts. We consider the GHFE which is the spin equivalent of the Zakharov-Ito equation (ZIE). We have established that these equations are gauge and geometrical equivalent to each other. The integrable motion of space curves induced by the ZIE is constructed. The 1-soliton solution of the GHFE is obtained from the seed solution of the ZIE.