| Abstract: |
| Schr\odinger models can be used to modelize different physical phenomena in many situations: self-channeling of a high-power ultra short laser in matter, dissipative quantum mechanics, plasma physics and fluid mechanics, propagation in quadratic media, ... However, little is known about Cauchy problem and the question of global well-posedness is still an open problem in many cases. In this direction, solitary waves and their stability properties is of great interest. In this talk, we propose to investigate two different types of models : a quasilinear Schr\odinger equation and a system of semilinear equations. Our aim is to prove the stability or instability of ground states, to exhibit the specificity of each model and to discuss some open problems. In particular, concerning solitonic structure arising in quadratic media, we will present, in the context of normal or anomalous dispersion regimes, the case of elliptic and nonelliptic systems. |
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