Darboux transformation and conservation laws for an inhomogeneous fifth-order nonlinear Schrödinger equation from the Heisenberg ferromagnetism

• A fifth-order nonlinear Schrödinger equation with variable coefficients is studied through Darboux transformation.
• The expressions of N-soliton solutions are derived.
• Conservation laws are constructed to confirm the integrability of the equation.
• We analyze the influence of perturbation terms and inhomogeneous parameters on the soliton propagation and interactions.

In this paper, we consider an inhomogeneous fifth-order nonlinear Schrödinger equation with variable coefficients, which describes the dynamics of a site-dependent Heisenberg ferromagnetic spin chain. Darboux transformation is constructed, and one- and two-soliton solutions are represented. In virtue of the Lax pair, we give an infinite number of conservation laws. Furthermore, we graphically analyze the influence of perturbation terms and inhomogeneous parameters on the soliton propagation and interactions. The perturbation terms are found to induce a phase shift of the soliton but do not affect the soliton amplitude. The inhomogeneous parameters lead to an increasing–decreasing process of soliton amplitude and the change of the propagation direction. Perturbation terms and inhomogeneous parameters also affect the soliton interactions.