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

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

Zbl 1072.35167
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