Mizumachi, Tetsu Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential. (English) Zbl 1146.35085 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. Cited in 19 Documents 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 PDF BibTeX XML Cite \textit{T. Mizumachi}, J. Math. Kyoto Univ. 47, No. 3, 599--620 (2007; Zbl 1146.35085) Full Text: DOI arXiv OpenURL