High frequency solutions of the nonlinear Schrödinger equation on surfaces

Abstract

Abstract. We address the problem of describing solutions of the nonlinear Schrödin-ger equation on a compact surface in the high frequency regime. In this context, we introduce a nonnegative threshold, depending on the geometry of the surface, which can be seen as a measurement of the nonlinear character of the equation, and we compute this number for the torus and for the sphere, as a consequence of earlier arguments. The last part is devoted to the study, on the sphere, of the critical regime associated to this threshold. We prove that the effective dynamics are described by a new evolution equation, the Resonant Hermite-Schrödinger equation.

The cubic Szegő equation

“…Moreover, a theorem by Peller states that, for p < ∞, the Schatten norm [Tr(|H u | p )] 1/p is equivalent to the norm of u in the Besov space B 1/p p,p , which is therefore uniformly bounded for all time if it is finite at t = 0. Notice that the particular case p = 2 was already observed in (10), giving again the conservation of M(u)+Q(u). Another example of a conserved quantity is of course the trace norm T r(|H u |) , which, as stated before, is equivalent to the Besov B 1 1,1 norm of u (or to the L 1 -norm of u ′′ with respect to the area measure in the disc).…”

“…Therefore a better understanding of equation ( 3) requires to study the interaction between the nonlinearity |u| 2 u and the projectors Π ± m . Notice that similar interactions arise in the literature, see for instance [22] in the study of the Lowest Landau Level for Bose-Einstein condensates, or [10] in the study of critical high frequency regimes of NLS on the sphere. Other examples can be found in the introduction of [13].…”