Modulational Instability and Location Controllable Lump Solutions with Mixed Interaction Phenomena for the (2+1)-Dimensional Myrzakulov-Lakshmanan-IV Equation

Abstract

AbstractUnder consideration is a $$(2+1)$$ ( 2 + 1 ) -dimensional spin nonisospectral Myrzakulov-Lakshmanan-IV (ML-IV) equation, which has close relation with the celebrated nonlinear Schrödinger equation. In the first place we study the modulational instability (MI) for this equation and deduce its formation mechanism for diverse localized waves from the plane wave background. Secondly, in view of the known nonisospectral Lax pair of this equation, we first successfully extend the generalized $$(n, N-n)$$ ( n , N - n ) -fold Darboux Transformation (DT) from $$(1+1)$$ ( 1 + 1 ) -dimensional equation to this $$(2+1)$$ ( 2 + 1 ) -dimensional equation. As an application of the resulting DT, we derive some location controllable dark and bright lump, periodic wave and mixed breather-lump solutions, it is shown that the position of these localized waves can be controlled by some special parameters, so that we can move them to the positions we want. Especially, the dynamical behaviors of these localized waves are illustrated graphically. These results may have potential value for the study of applied ferromagnetism and nano magnetism.