In this paper, an introduction to the new subject of nonlinear dispersive Hamiltonian
equations on graphs is given. The focus is on recently established properties of solutions in …

This textbook introduces the well-posedness theory for initial-value problems of nonlinear,
dispersive partial differential equations, with special focus on two key models, the Korteweg …

We define the Schrödinger equation with focusing, cubic nonlinearity on one-vertex graphs.
We prove global well-posedness in the energy domain and conservation laws for some self …

The theory of linear dispersive equations predicts that waves should spread out and
disperse over time. However, it is a remarkable phenomenon, observed both in theory and …

We use an effective one-dimensional Gross-Pitaevskii equation to study bright matter-wave
solitons held in a tightly confining toroidal trap** potential, in a rotating frame of reference …

We study the nonlinear Schrödinger equation with a delta-function impurity in one space
dimension. Local well-posedness is verified for the Cauchy problem in H1 (R). In case of …

We study existence and stability properties of ground-state standing waves for two-
dimensional nonlinear Schrödinger equation with a point interaction and a focusing power …

We study analytically and numerically the stability of the standing waves for a nonlinear
Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is …

Effective integrable dynamics for a certain nonlinear wave equation Page 255 ANALYSIS AND
PDE Vol. 5, No. 5, 2012 dx. doi. org/10.2140/apde. 2012.5. 1139 msp EFFECTIVE INTEGRABLE …

By introducing PT-symmetric δ-function potentials into three-component Gross–Pitaevskii
equations that describe spinor F= 1 Bose–Einstein condensates, we obtain stable and …