Hayato Miyazaki -- Asymptotic behavior of solutions to the long-range nonlinear Schrodinger equation on a star graph
Author information
- Hayato Miyazaki
- Kagawa University
- Website: http://www.ed.kagawa-u.ac.jp/~miyazaki/cv_eng.html
Abstract:
In this talk, we consider the nonlinear Schr"odinger equation with power-type long-range nonlinearities on a star graph with the Kirchhoff boundary condition. When the nonlinearity is critical in the sense of long time behavior, we prove that there exists a solution which asymptotically behaves like a free solution with a logarithmic phase correction as time goes on. This phenomena is called long-range scattering. Moreover, we also show the nonexistence of solutions to the equation which behave like the free solution without the phase correction. These results are extension of the results for the equation on the real line into the star graph. A new ingredient is a factorization property of the Schr"odinger operator on the star graph. To give the factorization property, we employ a generalized Fourier transform with respect to a Laplacian on the star graph introduced by Weder (2015).