Modified non linear Shrödinger equationsnew results motivated by applications

Abstract

The nonlinear Schrödinger equation is an example of a universal nonlinear model that describes many physical nonlinear systems. This equation !nds its way in various areas of nonlinear science such as nonlinear optics, hydrodynamics, nonlinear acoustics, it can also describe quantum condensates, propagation of heat pulses in solids, etc. The NLSE provides a canonical description of the envelope dynamics of a quasi-monochromatic plane wave (the carrying wave) propagating in a weakly nonlinear dispersive medium when dissipative processes are negligible. For short times and small propagation distances, the dynamics is linear, but nonlinear interactions result in a signi!cant modulation and can produce essentially nonlinear cumulative effects such as e.g. appearance of stable structures of solitonic type. In optics, it can also be viewed as the extension to nonlinear media of the paraxial approximations widely used for linear waves propagating in random media. The name "NLSE" originates from a formal analogy with the Schrödinger equation of quantum mechanics. In this context a nonlinear potential arises in the "mean !eld" description of interacting particles. When the NLSE is considered in the wave context, the secondorder linear operator describes the dispersion and diffraction of the wave-packet and the nonlinearity arises from the sensitivity of the refractive index of the medium to the wave amplitude. This thesis is devoted to the study of various problems closely associated with the NLSE such as the Zakharov system, the generalized Zakharov system, the system of coupled NLSEs, the stationary cubic--quintic NLSE and other highly nonlinear generalizations such as the $p(x)$--Laplacian equations.