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This paper explores the pure‐cubic nonlinear Schrödinger equation (PC‐NLSE) with different nonlinearities. According to qualitative analysis, we get the dynamic systems and show that solitons and periodic solutions exist. The corresponding traveling wave solutions of these equations are constructed to demonstrate the correctness of qualitative analysis, and some solutions are initially given. In particular, a special kind of soliton solution, the Gaussian soliton, is constructed, which is rarely identified in non‐logarithmic equation. Next, the solitons stability and modulation instability (MI) of PC‐NLSE with two types of nonlinearity are discussed. Finally, by adding perturbed terms to the dynamic systems, we obtain the largest Lyapunov exponents and the phase diagrams of the equation, which proves there are the chaotic behaviors in PC‐NLSE. To the best of our knowledge, the Gaussian solitons, stability analysis and chaotic behaviors we obtained are first presented, which improves the study and proposes a new direction for the future researches on PC‐NLSE.
Li et al. (2024) studied this question.