Bifurcations and Exact Solutions in a Nonlinear Schrödinger Equation With High-Order Nonlinearity

This paper investigates the dynamics of a nonlinear Schrödinger equation (NLSE) featuring seventh-order nonlinearity and weak nonlocality. By reducing the governing equation to a planar Hamiltonian system, we perform a systematic qualitative analysis and provide a complete topological classification of its phase portraits. In contrast to previous studies that derived specific chirped solitary waves using ansatz-based methods, our approach employs the bifurcation theory of dynamical systems to uncover the full spectrum of more than 42 distinct exact bounded solutions. These include not only smooth periodic and solitary waves but also non-smooth localized structures such as peakons, periodic peakons, and compactons. We provide explicit parametric representations for each wave profile and establish the parametric conditions under which they emerge. Our results provide a comprehensive map of the localized structures admitted by this high-order NLSE model.