WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity

Abstract

Abstract. We consider the semi-classical limit of nonlinear Schrödinger equations in the presence of both a polynomial nonlinearity and the derivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolic structure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time dependent analytic regularity.

WKB analysis of the Logarithmic Nonlinear Schrodinger Equation in an analytic framework

“…These assumptions are made to ensure hyperbolicity, but have the strong drawback to involve the solution itself. However, hyperbolicity is not needed when one works with analytic functions ( [12]). In this context, the semiclassical limit for (1.8) (with κ = 0) was studied by [29,44], thanks to some tools developed by J. Sjöstrand [42].…”

“…In this section, we prove Theorem 1.4. Our proof is based on an iterative scheme in a similar way as in [12] for example even though it is a little different.…”

“…The following lemma, based on the previous properties, is a toolbox for all the forthcoming analysis and estimates. For a partial proof, we refer to [12], most cases not treated in there can be treated in a similar way thanks to Lemma 2.1.…”