Bifurcations Analysis and Monotonicity of the Period Function of the Lakshmanan–Porsezian–Daniel Equation with Kerr Law of Nonlinearity

The bifurcations and monotonicity of the period function of the Lakshmanan–Porsezian–Daniel equation with Kerr law of nonlinearity are discussed. Firstly, by the traveling wave transformations, the Lakshmanan–Porsezian–Daniel equation is reduced to the planar Hamiltonian system whose Hamiltonian function includes a 6-th degree polynomial. Then we give the phase portraits of the Hamiltonian system, and some traveling waves including dark wave solutions, kink and anti-kink solutions and periodic solutions are constructed by using the bifurcation method of dynamical systems. Furthermore, we discuss the monotonicity of the period function of periodic wave solutions by using some Lemmas proposed by Yang and Zeng (Bull Sci Math 133(6):555-557, 2009). Finally, some numerical simulations are presented.