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On the stability of solitary waves for nonlinear Schrödinger equations. (English) Zbl 0841.35108

Uraltseva, N. N. (ed.), Nonlinear evolution equations. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 164 (22), 75-98 (1995).
The paper is devoted to the stability problem for the nonlinear Schrödinger equation \[ i\vec\psi_t= [- \partial^2_x+ V(\psi_1 \psi_2)] \sigma_3 \vec\psi, \] where \(\vec\psi={\psi_1(x,t)\choose\psi_2(x,t)}\), \(\sigma_3=\left(\begin{smallmatrix} 1 & 0\\ 0 & -1\end{smallmatrix}\right)\) and \(V\) is a real-valued function with \[ \begin{aligned} V(\xi) & \geq - V_1 \xi^q,\quad V_1> 0,\quad \xi\geq 1,\quad q< 2,\\ V(\xi) & = V_2 \xi^p(1+ O(\xi)),\quad p> 0,\quad \xi\to 0.\end{aligned} \] {}.
For the entire collection see [Zbl 0824.00037].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
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