Construction and exact solution of the nonlocal Kuralay-II equation via Darboux transformation

Abstract

The Kuralay-II equation, as a typical form of the well-known Heisenberg ferromagnet equation, is an important integrable model. Here, the nonlocal Kuralay-II equation is constructed for the first time by means of symmetry reduction, resulting in an integrable system of partial differential equations. To solve this equation, Darboux transformation method is employed, which transforms the equation form to eliminate the influence of spectral parameters in the denominator and constructs a suitable gauge transformation matrix. Using trivial solutions as seed solutions, exact solutions of the equation are obtained, and the parameter constraint relationships when spectral parameters take real numbers, conjugate complex numbers, and unrelated complex numbers are analyzed, with specific examples given for the first two cases. This research contributes to solving nonlocal partial differential equations and enriches the construction methods of exact solutions in soliton theory.