Remarks on the global dynamics for solutions with an infinite group invariance to the nonlinear Schrödinger equation (Harmonic Analysis and Nonlinear Partial Differential Equations)

"Harmonic Analysis and Nonlinear Partial Differential Equations". July 3～5, 2017. edited by Hideo Takaoka and Hideo Kubo. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We consider the focusing mass-supercritical and energy-subcritical nonlinear Schrödinger equation (NLS). The global dynamics below the ground state standing waves is known (see [6, 1, 9]). Recently, the author [12] gave the global dynamics above the ground state standing waves for finite group invariant solutions. In the present paper, we are interested in the global dynamics for the solutions with an infinite group invariance

Two-solitons with logarithmic separation for 1D NLS with repulsive delta potential

We consider the one-dimensional nonlinear Schr\"odinger equation with focusing, power nonlinearity, and a repulsive delta potential. We show that if the potential is not too strong, the construction by Nguy\~{\^e}n (2019) of solutions converging strongly at time infinity to a pair of logarithmically separating solitons can be adapted to accommodate the effect of the potential. On the other hand, we show that if the potential is stronger, no such solutions exist