Generation of optical solitons molecules and energy flow in Painlevé-integrable Schrödinger dynamical systems

We study the perturbed and modified nonlinear Schrödinger equation (PNLSE with Kerr nonlinearity and MNLSE), establish their integrability under explicit parameter conditions via the Painlevé test, and construct analytical traveling-wave families. Methodologically, we (i) prove integrability through a complete dominant-balance and resonance analysis, (ii) derive closed-form polynomial/elliptic solutions with the Unified Method (UM) for both models, and (iii) obtain periodic solutions using the Energy Balance Method (EBM). The novelty is the combined, comparative use of UM and EBM on these two optical models, with parameter maps linking coefficients to solution classes (solitary, soliton/dromion, and elliptic/periodic profiles). We also interpret the physical relevance for nonlinear optics (optical pulse propagation with Kerr response, self-steepening) and compare our exact profiles with prior solution families. These results provide verifiable benchmarks for NLSE-type dynamics and broaden the toolbox for modeling optical solitons and periodic waves in Kerr media.