Bifurcations and Traveling Wave Solutions of Lakshmanan–Porsezian–Daniel Equation with Parabolic Law Nonlinearity

Abstract

In this paper, we investigate the bifurcations and exact traveling wave solutions of the Lakshmanan–Porsezian–Daniel model with parabolic law nonlinearity. Employing the bifurcation theory of planar dynamical system, we obtain the phase portraits of the corresponding traveling wave system. The existence of the two singular straight lines leads to the dynamical behavior of solutions with two scales. Corresponding to some special level curves, we give the explicit exact solutions under different parameter conditions, containing peakon solutions, solitary wave solutions, kink and anti-kink wave solutions, periodic wave solutions and periodic peakon solutions. Moreover, we collect all these traveling solutions into a theorem. Finally, we discuss a small perturbation of the LPD equation and persistence of the traveling waves.