Chaotic Phenomenon, Variational Principle, Hamiltonian, Phase Portraits, Bifurcation Analysis and Diverse Soliton Solutions of the Unstable Schrödinger Equation Involving Kerr Law in Optical Fibers

The unstable Schrödinger equation involving Kerr law in optical fibers is explored quantitatively and qualitatively in this work. With the aid of the semi-inverse method (SIM) and traveling wave transformation, we develop the variational principle (VP). The corresponding Hamiltonian function is extracted based on the VP. Make using of the Galilean transformation, we derive the planar dynamical system. Then the phase portraits are presented and bifurcation analysis is given to discuss the existence conditions of the different shape wave solutions. Furthermore, the chaotic phenomenon is also probed through introducing the external disturbances. In the end, three powerful techniques, the invariant algebraic curve approach that is based on the planar dynamical system, the Hamiltonian-based method which is on the basis of the energy conservation as well as the Wang’s direct mapping method, are used to construct the diverse wave solutions, which include the bell shape soliton, anti-bell shape soliton, singular wave, singular periodic wave and periodic wave solutions. Meanwhile, The outlines of the extracted wave solutions are unfolded with the help of Maple software to show the physical behaviors. As expected, some exact solutions extracted by Wang’s direct mapping method are the same with that obtained through the invariant algebraic curve approach. This strongly shows the correctness of the mentioned methods. To the best of the author's knowledge, the qualitative analysis for the problem is presented for the first time. Moreover, the methods used in this study are more simple and direct, which can avoid a lot of tedious calculations.