Soliton Outcomes and Stability Study of the Schrödinger Equation With Cubic Nonlinearity

This is the authors' abstract. We don't add key points for this paper.

This article presents novel soliton solutions for the nonlinear Schrödinger equation with cubic nonlinearity, which arises in quantum mechanics and nonlinear optics. The ordinary differential framework of this model is obtained using a traveling wave transformation. We employ the unified, improved Kudryashov, and generalized Kudryashov methods to derive exact analytical solutions. These include rational, hyperbolic, exponential, and trigonometric forms, yielding diverse soliton structures such as bright-dark breathers, multiple breathers, localized breathers, periodic waves, double-periodic waves, kinks, and antikinks. The physical behavior of these solutions is illustrated graphically. A Hamiltonian analysis is used to examine the energy conservation and the stability of the solution, and the stability of equilibrium points is studied under various parameters. Our results align with previous findings and may be useful in understanding nonlinear wave behavior in physics, engineering, and applied mathematics.