Section: New Results

Schrödinger equations

S. Rota Nodari, G. Dujardin, S. De Bièvre and collaborators continued their previous work on periodically modulated optical fibers with the experimental physicists of PhLAM [19]. They show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes.

In [42] S. de Bièvre and G. Dujardin analyze the formation of the Kuznetsov-Ma soliton of the 1D Schrödinger equation in the presence of periodic modulation satisfying an integrability condition. They show that this particular soliton has several compression points, the number, position and shape of which are controlled by the amplitude and the frequency of the modulation. They analyze the interplay between the frenquency of the soliton and the frequency of the modulation. Moreover, they show that one can suppress any component of the output spectrum of the soliton by a suitable choice of the amplitude and frequency of the modulation.

These works are part of the activities developped in the LabEx CEMPI.

In [54], D. Bonheure, J.-B. Casteras and R. Nascimento obtained new results on the existence and qualitative properties of waveguides for a mixed-diffusion NLS. In particular, they proved the first existence results for waveguides with fixed mass and provided several qualitative descriptions of these.

S. De Bièvre and S. Rota Nodari continued their work on orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces, with a second paper [37], dealing with the situation where multi-dimensional invariance groups are present in the systems considered. They present a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and show how it allows to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method is applied to study the orbital stability of the plane waves of a system of two coupled nonlinear Schrödinger equations. They provide a comparison of their approach to the classical one by Grillakis-Shatah-Strauss.