$N$-soliton solutions, Bäcklund transformation and conservation laws for the integro-differential nonlinear Schrödinger equation from the isotropic inhomogeneous Heisenberg spin magnetic chain

Under investigation in this paper is an integro-differential nonlinear Schrödinger (IDNLS) equation, which is equivalent to the spin evolution equation of a classical in-homogeneous Heisenberg magnetic chain in the continuum limit. Based on the Hirota method, the bilinear form and $N$-soliton solution for the IDNLS equation are derived with the help of symbolic computation. Moreover, $N$-soliton solution for the IDNLS equation is expressed in terms of the double Wronskian and testified through the direct substitution into the bilinear form. Besides, the bilinear Bäcklund transformation and infinitely many conservation laws are also obtained for the IDNLS equation. Propagation characteristics and interaction behaviors of the solitons are discussed by analysis of such physical quantities as the soliton amplitude, width, velocity and initial phase. Interactions of the solitons are proved to be elastic through the asymptotic analysis. Effect of inhomogeneity on the interaction of the solitons is studied graphically.