Resonant nonlinear light propagation via optical waveguides with localized flaws and general geometric interpretation

This paper offers a comprehensive analytical exploration of the nonlinear Schrödinger equation (NLSE), focusing on its symmetrical properties, invariant configurations, and wave propagation solutions. Using the classical Lie group technique, we find the full set of infinitesimal symmetry generators that the nonlinear Schrödinger equation allows. A complete system of one-dimensional subalgebras is then created, which makes it easier to classify symmetry reductions and find invariant solutions. From the symmetries that have been identified, conservation laws are derived, which lead to conserved quantities that have physical significance, such as mass, momentum, and energy. Additionally, by utilising an effective integration method known as the extended rational sinh-Gordon equation approach (ERSGEA), several types of exact solitons, including mixed rational dark-bright, mixed rational singular, single rational dark, and single rational singular solutions in the form of hyperbolic functions, are systematically created and analysed. The results provide a more thorough knowledge of the NLSE’s integrable structure and its complex solution dynamics, which have direct effects on nonlinear optics, plasma physics, and Bose–Einstein condensates.