Exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity by using the first integral method

The nonlinear equations of mathematical physics are major subjects in physical science [1]. Recently many new approaches for finding the exact solutions to nonlinear wave equations have been proposed, for example, tanh-sech method [2–4], extended tanh method [5–7], sine-cosine method [8–10], homogeneous balance method [11, 12], Jacobi elliptic function method [13–16], F -expansion method [17–19], exp-function method [20, 21], trigonometric function series method [22], (G′/G)-expansion method [23, 24] and so on. All methods mentioned above have limitation in their applications. The first integral method was first proposed by Feng [25] in solving Burgers-KdV equation which is based on the ring theory of commutative algebra. Recently, this useful method is widely used by many such as in [26, 27] and by the reference therein. In this paper, we will consider the perturbed NLSE with Kerr law nonlinearity [28] with following form: