"Many physical models involve nonlinear dispersive problems, like waveor laser propagation, plasmas, ferromagnetism, etc. So far, the mathematical under-standing of these equations is rather poor. In particular, we know little about thedetailed qualitative behavior of their solutions. Our point is that an apparent com-plexity hides universal properties of these models; investigating and uncovering suchproperties has started only recently. More than the equations themselves, these univer-sal properties are essential for physical modelisation.By considering several standard models such as the nonlinear Schrodinger, nonlinearwave, generalized KdV equations and related geometric problems, the goal of this pro-posal is to describe the generic global behavior of the solutions and the profiles whichemerge either for large time or by concentration due to strong nonlinear effects, if pos-sible through a few relevant solutions (sometimes explicit solutions, like solitons). Inorder to do this, we have to elaborate different mathematical tools depending on thecontext and the specificity of the problems. Particular emphasis will be placed on- large time asymptotics for global solutions, decomposition of generic solutions intosums of decoupled solitons in non integrable situations,- description of critical phenomenon for blow up in the Hamiltonian situation, stableor generic behavior for blow up on critical dynamics, various relevant regularisations ofthe problem,- global existence for defocusing supercritical problems and blow up dynamics in thefocusing cases.We believe that the PI and his team have the ability to tackle these problems at present.The proposal will open whole fields of investigation in Partial Differential Equations inthe future, clarify and simplify our knowledge on the dynamical behavior of solutionsof these problems and provide Physicists some new insight on these models."