Soliton and rogue-wave solutions for a (2 + 1)-dimensional fourth-order nonlinear Schrödinger 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 Schrödinger 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 N th-order rogue-wave solutions via the generalized DT. Profiles for the solitons and rogue waves are analyzed with respect to the lattice parameter σ , respectively. When σ is greater than a certain value marked as σ _0 , one-soliton velocities increase with the increase of σ . When σ <σ _0 , one-soliton velocities decrease with the increase of σ . When the time t is equal to zero, σ has no effect on the interactions between the two solitons. When t 0 , different choices of σ 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 σ , while the amplitudes do not depend on σ . The second-order rogue waves are composed by three first-order rogue waves whose widths all get wider with the decrease of σ , while the amplitudes do not depend on σ .