Localized Coherent Structures of Ishimori Equation I through Hirota's Bilinearization method:Time dependent/Stationary boundaries

Abstract

Ishimori equation is a $(2+1)$ dimensional generalization of the $(1+1)$ dimensional integrable classical continuous Heisenberg ferromagnetic spin equation. The richness of the coherent structures admitted by Ishimori equation I such as dromion, lump and rationally- exponentially localized solutions, have been demonstrated in the literature through $\bar \partial$ technique and binary Darboux transformation method. To our knowledge Hirota's method had been adopted to construct only the vortex solutions of Ishimori equation II. For the first time, the various types of localized coherent structures mentioned above have been constructed in this paper for the Ishimori equation I using the Hirota's direct method. In particular we have brought out the significance of boundaries and arbitrary functions in generating all these types of localized structures and proved that the absence of such boundaries leads only to line soliton solutions.