<p>The Lakshmanan-Porsezian-Daniel system with Kerr law of nonlinear refractive index is a traditional model which has been wildly applied in numerous fields, circulation system of chemical industry, plasma physics, optical-fiber communications, nonlinear optics, fluid mechanics and heat pulses in solids. With the help of the second-order and triple-order polynomial complete discriminant system method, we obtain a series of new traveling wave solutions, such as Jacobian elliptic function solutions, solitary wave solutions, singular soliton solutions, bright soliton solutions, dark soliton solutions, triangular function solutions and rational function solutions. To be honest, we give all the classification of single traveling wave solutions for the Lakshmanan-Porsezian-Daniel system. The obtained solutions may improve or complement previous results. Hence, the result has practical significance for the study of wave propagation in the field of optical-fiber communications.</p>