Periodic solutions and Whitham modulation equations for the Lakshmanan–Porsezian–Daniel equation

In this paper, we adopt the improved finite-band integration method to find the periodic solutions of the Lakshmanan-Porsezian-Daniel (LPD) equation, which can be used to describe the nonlinear spin excitation with octupole-dipole interaction in the one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin chain. Based on this method, we obtain the one-phase periodic solutions of the LPD equation represented by the Weierstrass elliptic functions and Jacobi elliptic functions. Furthermore, we derive the Whitham velocities and Whitham modulation equations of the LPD equation based on the Riemann invariants.