Modified scattering for odd solutions of cubic nonlinear Schrödinger equations with potential in dimension one

Jean-Marc Delort (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

We show that the global odd solutions of a cubic Schrödinger equation with potential, with small smooth decaying initial data, do not scatter in one space dimension. More precisely, we obtain for the asymptotics of such solutions an explicit expression, involving a logarithmic modulation in the phase of oscillation. This property has been known for long in the potentialless case. In the presence of a (generic) potential, some commutation issues of the Klainerman vector field like operator used in order to exploit dispersion appear. Our method of proof uses the wave operators of the stationary Schrödinger operator, in order to reduce the problem to an equation without potential, but with a variable coefficients pseudodifferential nonlinearity. Exploiting the fact that we are working only with odd solutions, we may overcome the commutation issues alluded to above, and, using semiclassical analysis, deduce from the PDE an ODE, whose analysis provides the wanted asymptotics of the solution.

Mots clés: Cubic Schrödinger equation; Modified scattering; Wave operators

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Marc Delort. Modified scattering for odd solutions of cubic nonlinear Schrödinger equations with potential in dimension one. 2016. ⟨hal-01396705⟩