Bifurcation, chaos, modulation instability, and soliton analysis of the schrödinger equation with cubic nonlinearity

Bifurcation, chaos, modulation instability, and solitons are important phenomena in nonlinear dynamical structures that help us understand complex physical processes. This work employs the Schrödinger equation with cubic nonlinearity (SECN), rising in superconductivity, quantum mechanics, optics, and plasma physics. We developed an ordinary differential arrangement using a traveling wave alteration and found the soliton outcome to the mentioned problem utilizing the unified solver technique. Then we obtain different categories of wave designs, including bright and dark solitons, with periodic, quasiperiodic, and chaotic behaviors. We examined the system's planar dynamics to observe sensitivity, chaos, and bifurcation. The chaotic state involves various procedures such as multistability, recurrence diagrams, bifurcation shapes, fractal dimensions, return maps, Poincaré graphics, Lyapunov exponents, phase portraits, and strange attractors. These properties help us understand complicated behaviors like how waves become chaotic patterns. Our outcomes demonstrate a clear understanding of wave propagation and how energy is absorbed. These findings develop our knowledge of nonlinear problems and guide us in wave circulation in real-life situations. Therefore, our procedures are useful and operative offering a simple technique for studying nonlinear problems, which progresses our understanding of these equations' sensitivity, waveform pattern, and propagation strategy.