Étude de quelques propriétés d'équations d'ondes non linéaires dispersives de type Schrödinger
Thèses d'Orsay, no. 304 (1992) , 102 p.

This thesis is divided into three parts, each one featuring the study of some properties of nonlinear Schrödinger type dispersive wave equations arising in several fields of physics.

In the first part, we study the Cauchy problem associated with a nonlinear Schrödinger equation with an external magnetic field. Under some growth restrictions on the potentials and the nonlinear term in the equation, we prove the local existence and the uniqueness of solutions for the Cauchy problem for this equation in a weighted Sobolev space. We also prove the conservation of the associated energy.

In the second part, we study the existence of smooth analytic solutions for a general nonlinear Schrödinger type equation. This equation contains some physical models arising in the context of water waves. In these models, the linear term may be a differential operator of order larger than two, and the nonlinear term may be nonlocal.

The third part is devoted to the study of the existence and the instability of some localised stationary solutions of a nonlinear Schrödinger equation with a general nonlinearity. These localised solutions have a nonzero limit when the space variable goes to infinity, and for some particular nonlinear terms, they have a definite physical interpretation. We prove, by linearizing the equation, that when these solutions exist, they are always unstable solutions of the evolution equation.

Mots clés : Schrödinger equation, external magnetic fields, water waves, solitary waves, orbital stability 
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     author = {De Bouard, Anne},
     title = {\'Etude de quelques propri\'et\'es d'\'equations d'ondes non lin\'eaires dispersives de type {Schr\"odinger}},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e de Paris-Sud Centre d'Orsay},
     number = {304},
     year = {1992},
     language = {fr},
     url = {http://archive.numdam.org/item/BJHTUP11_1992__0304__P0_0/}
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De Bouard, Anne. Étude de quelques propriétés d'équations d'ondes non linéaires dispersives de type Schrödinger. Thèses d'Orsay, no. 304 (1992), 102 p. http://numdam.org/item/BJHTUP11_1992__0304__P0_0/

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