Mathematics

In this talk we will discuss  some joint work with Masaya Maeda (Chiba University)  about the asymptotic stability  of ground states of the  pure power Nonlinear Schrödinger Equation  (NLS)     in space dimension 1  for a certain range of exponents.  T. Cazenave and and P.L. Lions proved the orbital stability in 1982 but the issue of stability  is  much older. For example,  the Vakhitov-Kolokolov's stability criterion was published in 1973. A  non rigorous description of the spectral stability can be found  in the series of textbooks by Landau and Lifsits. We will present the problem, which in many respects is still open, and  is  completely open in all higher dimensions. In the problem there is a mixture of discrete and continuous components. The discrete components lose energy because of nonlinear interaction with the continuous component. We will brieflymentions this phenomenon which goes under the name of Nonlinear Fermi Golden Rule. As it spills in the continuous component, the energy scatters mainly because of linear dispersion. However, since the system is nonlinear, it is potentially very difficult to prove dispersion. In fact, some of the most important contributions in the theory of Nonlinear  Wave type Equations (to which NLS belongs)  has focused on this problem (the Klainerman vector fields; the Christodoulou compactification; the Shatah normal forms; the Germain,Masmoudi  and Shatah theory of resonances).   However here we use an approach developed by Merle and his school, which has led to some of the most important results on dispersive equations in the last 20 year (his work with Martel on KdV; his work with Raphael on the blow up for the NLS). This involves the use of Virial Inequalities which, at least in dimension 1,  overcome very simply hurdles which with other methodologies are very  or impossibly difficult. This was used also recently by Martel   who recently solved an asymptotic stability problem for small amplitude  ground states of the NLS with nonlinearity which is a sum of a cubic and a quantic power in 1 dimension. However, Martel's technique works only for small amplitude ground states. We have a modification of the method which seems very promising, where we couple a simple high energy  virial inequality with the classical Kato smoothing. In the talk I will try to frame the problem, give a sense of the difficulties and give a very sketchy desxription of the structure of the proof.