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

EPFL 01/2008; DOI: 10.5075/epfl-thesis-4233
Source: OAI

ABSTRACT

The nonlinear Schrödinger equation (1) i∂tw + Δw +V(x)|w|p-1w = 0 w = w(t,x) : Ι × R^N → C, N ≥ 2, is studied, with p > 1, V : R^N \ {0} → R and Ι ⊂ R an interval. The coefficient V is subject to various hypotheses. In particular, it is always assumed that V (x) → 0 as |x| → ∞. Situations where V is unbounded at the origin are considered. A special attention is paid to the radial case. Seeking solutions of (1) as standing waves w(t,x) = e^{iλt}u(x) leads naturally to the semilinear elliptic equation (2) Δu - λu + V(x)|u|^{p-1}u = 0 u : R^N → R, N ≥ 2. The main goals of the thesis are to establish existence and bifurcation results for (2), to discuss the orbital stability of the standing waves of (1) corresponding to the solutions found in (A). First, in Chapter 1, in the case where V is radial, a variational approach shows the existence of ground states for (2). A non-degeneracy property of these solutions is proved, which plays a crucial role in the continuation arguments of Chapter 2. The first part of Chapter 2 establishes local existence and bifurcation results for (2), without any symmetry assumption on V . Under certain hypotheses on the power p and the coefficient V , two branches of solutions are obtained, in a neighborhood of λ = 0 and in a neighborhood of λ = +∞. The branches are of class C^r if V ∈ C^r(R^N \ {0},R), for r = 0, 1. These independent results are proved by requiring respectively that lim_{|x|→∞} V (x)|x|^b = B > 0 with b ∈ (0, 2) and that lim_{x→0} V (x)|x|^a = A > 0 with a ∈ (0, 2). The asymptotic behaviour along the branches is discussed in detail and depends on the value of p. The second part of Chapter 2 proves the existence of a global branch of solutions of (2), in the case where V is radial. Under appropriate hypotheses, in particular if a ∈ (0, b], the global branch "sticks together" the two local branches obtained in the first part. Chapter 3 is concerned with the orbital stability of the standing waves of (1) corresponding to the solutions of (2) found in the first part of Chapter 2. It is explained in detail how to apply the general theory of orbital stability to (1). Local stability/instability results are proved, in a neighborhood of λ = 0 and in a neighborhood of λ = +∞.

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Available from: Francois Genoud
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• "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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• "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.
Preview · Article · Oct 2011