Gauge equivalent structures of the integrable (2+1)-dimensional nonlocal nonlinear Schrödinger equations and their applications

Abstract

In this paper, we are concerned with the gauge equivalent structures for the integrable (2+1)-dimensional nonlocal nonlinear Schrödinger (NLS) equations. Through constructing the gauge transformation, we prove that these (2+1)-dimensional nonlocal equations, both focusing and defocusing, are gauge equivalent to two types of coupled (2+1)-dimensional Heisenberg ferromagnet equations and two types of coupled (2+1)-dimensional modified Heisenberg ferromagnet equations. As an appropriate extension, we further illustrate that the nonlocal NLS equation is gauge equivalent to two types of coupled Heisenberg ferromagnet equations and two types of coupled modified Heisenberg ferromagnet equations, while its discrete version is gauge equivalent to two types of coupled discrete Heisenberg ferromagnet equations and two types of coupled discrete modified Heisenberg ferromagnet equations, respectively. From its invariance with the combined parity-reflection and time-reversal operators, we can observe that there exist significant differences and intimate connections between standard and nonlocal equations. On the other hand, by using the Darboux transformation and some limit techniques, two types of deformed soliton solutions, namely, the deformed exponential solitons and the deformed rational solitons for the (2+1)-dimensional nonlocal defocusing NLS equation are given explicitly. With no loss of generality, two deformed soliton interactions and their various degenerate cases are discussed and illustrated through some figures.