Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential. (English) Zbl 1146.35085

Mizumachi, Tetsu

J. Math. Kyoto Univ. 47, No. 3, 599-620 (2007).

Summary: We consider asymptotic stability of a small solitary wave to supercritical two-dimensional nonlinear Schrödinger equations
\[ iu_t+\Delta u=Vu\pm|u|^{p-1}u \quad\text{for }(x,t)\in\mathbb R^2\times\mathbb R, \]
in the energy class. This problem was studied by S. Gustafson, K. Nakanishi and T.-P. Tsai [Int. Math. Res. Not. 2004, No. 66, 3559–3584 (2004; Zbl 1072.35167)] in the \(n\)-dimensional case \((n\geq 3)\) by using the endpoint Strichartz estimate. Since the endpoint Strichartz estimate fails in two-dimensional case, we use a time-global local smoothing estimate of Kato type to prove the asymptotic stability of a solitary wave.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35Q51	Soliton equations
35B35	Stability in context of PDEs

Keywords:

nonlinear Schrödinger equations; small solitons; asymptotic stability; Strichartz estimate; estimate of Kato type

Citations:

Zbl 1072.35167