Odd-soliton solutions and inelastic interaction for the differential-difference nonlinear Schrödinger equation in nonlinear optics

Nonlinear Schrödinger equation-type may model diverse physical phenomena in nonlinear optics, plasma physics and fluid mechanics, etc. Under consideration in this paper is the differential-difference nonlinear Schrödinger equation. On the basis of its Lax pair, N-fold Darboux transformation and conservation laws for the differential-difference nonlinear Schrödinger equation are constructed. Odd-soliton solutions in terms of determinant are derived with the resulting Darboux transformation. Figures are plotted to reveal the dynamic features of the solitons. Especially, the inelastic interaction phenomena among the three solitons are discussed for the differential-difference nonlinear Schrödinger equation, which might be useful for understanding some physical phenomena in nonlinear optics.