We consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387–406.] , where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

By using reductive perturbation technique we have studied the linear and non-linear properties of ion-acoustic solitary structures in a three-component plasma containing non-thermal electrons and Boltzmann positrons and a comparatively cold ion which has got a streaming motion. The Korteweg–de Vries equation has been obtained and the dependence of small amplitude solitary structures on various plasma parameters such as streaming velocity (v0), non-thermal parameter (β), reciprocal of electron temperature (χ), positron density (p), Mach number (M), and ion density (δ) have been studied. The possibility of formation of enveloping soliton and its characteristic features are further investigated by deriving the non-linear Schrödinger equation.