Cascade of phase shifts for nonlinear Schrodinger equations
Remi Carles
TL;DR
The paper studies the semi-classical limit of a nonlinear Schrödinger equation with quadratically oscillatory initial data that focus at $t=1$, in a super-critical caustic regime where nonlinear effects enter at leading order in the phase. They formalize a cascade of boundary layers and phase shifts, with a first boundary layer producing a leading nonlinear phase $\phi_1$ within a layer of size $\varepsilon^\beta$, where $\beta=(\alpha-1)/(n\sigma-1)$, and, in the homogeneous cubic case ($\sigma=1$), an infinite sequence of phases at scales $\varepsilon^{(j\alpha-1)/(jn\sigma-1)}$. The core method is a rigorous analysis based on a semi-classical conformal transform to recast $u^\varepsilon$ into $\psi^\varepsilon$ and a Grenier’s WKB ansatz $\psi^\varepsilon = a^\varepsilon e^{i\phi^\varepsilon/\varepsilon}$, yielding a well-posed coupled system and convergence results as $\varepsilon\to0$, separating a near-caustic nonlinear regime from linear dynamics away from the focus. They show that after the cascade nonlinear geometric optics governs near $t=1-\varepsilon^\gamma$, and discuss stability issues and open questions about caustics and long-time dynamics.
Abstract
We consider a semi-classical nonlinear Schrodinger equation. For initial data causing focusing at one point in the linear case, we study a nonlinearity which is super-critical in terms of asymptotic effects near the caustic. We prove the existence of infinitely many phase shifts appearing at the approach of the critical time. This phenomenon is suggested by a formal computation. The rigorous proof shows a quantitatively different asymptotic behavior. We explain these aspects, and discuss some problems left open.