N -bright-dark-soliton, the general soliton molecules and N -breather for the Kuralay equation

Abstract As a representative of the celebrated Heisenberg ferromagnet-type system, the Kuralay equation is investigated by the Darboux transformation (DT) method. To establish the integrability of the system, we first derive its Lax pair and the corresponding N -fold DT in a compact determinant form. Beginning with the mixed backgrounds, the mixed multi-solitons are derived via the resulting DT, and these solutions comprise N -bright-soliton in q component and N -dark-soliton in v component. The general soliton molecules (SMs) are then constructed by applying the velocity resonance conditions to the aforementioned soliton solutions. Furthermore, the general SMs can be classified into three primary categories: pure SMs, SMs interacting with fundamental solitons, and hybrid solutions comprising different types of SMs. Beginning from the non-zero backgrounds, various types of N -breather solutions are also generated, such as Akhmediev breather, Kuznetsov-Ma breather and the general case. Finally, the dynamics of these three kinds of nonlinear localized waves are discussed in detail.