# Théorie de bifurcation et de stabilité pour une équation de Schrödinger avec une non-linéarité compacte

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Francois Genoud, Jul 08, 2015 Available from:-
- "The physical relevance of (1.1) with b > 0 may not appear obvious due to the singularity at x = 0. However, this model problem plays an important role as a limiting equation in the analysis of more general inhomogeneous problems of the form i∂ t u + ∆u + V (x)|u| p−1 u = 0 with V (x) ∼ |x| −b as |x| → ∞, which are ubiquitous in nonlinear optics — see [5] [6] [8] for more details. We consider here strong solutions u = u(t, x) ∈ C 0 t H 1 x ([0, T ) × R N ), where T > 0 is the maximum time of existence of u. "

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**ABSTRACT:**We establish the classification of minimal mass blow-up solutions of the $L^2$ critical inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^{\frac{4-2b}{N}}u = 0, \] thereby extending the celebrated result of Merle from the classic case $b=0$ to the case $0<b<\min\{2,N\}$, in any dimension $N\ge1$. -
- "Using the general theory of orbital stability of Grillakis, Shatah and Strauss [9], we obtained in [7] [4] [5] various stability/instability results for general nonlinearities V (x)|φ| 2σ φ by studying the monotonicity of the L 2 norm of the standing waves, as a function of ω > 0. It turned out that σ = (2 − b)/N is a threshold for stability in the regimes we considered. For this value of σ, we could not determine if the standing waves are stable or not, even in the model case V (x) = |x| −b . "

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**ABSTRACT:**An inhomogeneous nonlinear Schrödinger equation is considered, which is invariant under the L^2-scaling. The sharp condition for global existence of H^1-solutions is established, involving the L^2-norm of the ground state of the stationary equation. Strong instability of standing waves is proved by constructing self-similar solutions blowing up in finite time.Zeitschrift für Analysis und ihre Anwendungen = Journal of analysis and its applications 01/2012; 31(3):283-290. DOI:10.4171/ZAA/1460 · 0.70 Impact Factor -
- "Using the general theory of orbital stability of Grillakis, Shatah and Strauss [9], we obtained in [7] [4] [5] various stability/instability results for general nonlinearities V (x)|φ| 2σ φ by studying the monotonicity of the L 2 norm of the standing waves, as a function of ω > 0. It turned out that σ = (2 − b)/N is a threshold for stability in the regimes we considered. For this value of σ, we could not determine if the standing waves are stable or not, even in the model case V (x) = |x| −b . "

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**ABSTRACT:**An inhomogeneous nonlinear Schr\"odinger equation is considered, that is invariant under $L^2$ scaling. The sharp condition for global existence of $H^1$ solutions is established, involving the $L^2$ norm of the ground state of the stationary equation. Strong instability of standing waves is proved by constructing self-similar solutions blowing up in finite time.