A note of the Stability of Standing Waves for the Semilinear Schr?dinger Equations

Abstract

We study the chapter eight of the book ”Semilinear Schr?dinger equations” written by Professor Thierry Cazenave which is to discuss the existence and stability of standing wave solutions for a semilinear Schr?dinger equation First giving the expression of the standing wave and substituting the standing wave into the semilinear Schr?dinger equation the author got an elliptic equation Then derived the Pohozaev identity some other identities and some related functionals to prove that the minimizer of the functional under a suitable constraint condition which is the solution of the elliptic equation In addition the argument of the convergence of the minimizing sequence only requires weak convergence ! The family of ground states can be generated by the unique good function through the modulo space transformation and Rotation In the case of the dimension 1 every bound state is a ground state This is not true when the dimension is greater than 1 ! Second when the equation has strong non-linear interaction the author gives some lemmas and corollaries to show that the standing wave is unstable The strategies for the proof is to show that if the initial data is close to the ground state then the corresponding maximal solution shall blow up in finite time Furthermore when the equation has weak non-linear interaction the author use the Concentration Compactness principle to claim that the minimizing sequence strong converges in the energy space and the ground state is the minimizer of an appropriate minimization problem Therefore the standing wave solution is then proved by contradiction to be orbitally stable The last part is to prove that under a certain non-linear exponent the mass of the ground state is related to the best constant of the Gagliardo-Nirenberg inequality In addition to studying the contents of this chapter we elaborate the proof in details which omitted by the author and we corrected some minor errors and typos

Date of Award	2020
Original language	English
Supervisor	Yung-Fu Fang (Supervisor)