Soliton and rogue-wave solutions for a (2+1)-dimensional fourth-order nonlinear Schrodinger equation in a Heisenberg ferromagnetic spin chain

In this paper, we investigate the soliton and rogue-wave solutions for a (2 + 1)-dimensional fourth-order nonlinear Schrodinger equation, which describes the spin dynamics of a Heisenberg ferromagnetic spin chain with the bilinear and biquadratic interactions. For such an equation, there exists a gauge transformation which converts the nonzero potential Lax pair into some constant-coefficient differential equations. Solving those equations, vector solutions for the nonzero potential Lax pair are obtained. The condition for the modulation instability of the plane-wave solution is also given through the linear stability analysis. Then, we present the determinant representations for the N-soliton solutions via the Darboux transformation (DT) and Nth-order rogue-wave solutions via the generalized DT. Profiles for the solitons and rogue waves are analyzed with respect to the lattice parameter sigma, respectively. When sigma is greater than a certain value marked as sigma(0), one-soliton velocities increase with the increase of sigma. When sigma < sigma(0), one-soliton velocities decrease with the increase of sigma. When the time t is equal to zero, sigma has no effect on the interactions between the two solitons. When t not equal 0, different choices of sigma lead to the different two-soliton velocities, giving rise to the different interaction regions. Widths of the first-order rogue waves become bigger with the decrease of sigma, while the amplitudes do not depend on sigma. The second-order rogue waves are composed by three first-order rogue waves whose widths all get wider with the decrease of sigma, while the amplitudes do not depend on sigma.