Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1312.5467

Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field

Jonathan Di Cosmo·Jean Van Schaftingen
View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct solutions to the nonlinear magnetic Schrödinger equation $$ \left\{ \begin{aligned} - \varepsilon^2 Δ_{A/\varepsilon^2} u + V u &= \lvert u\rvert^{p-2} u & &\text{in}\ Ω,\\ u &= 0 & &\text{on}\ \partialΩ, \end{aligned} \right. $$ in the semiclassical régime with strong magnetic fields. In contrast with the well-studied mild magnetic field régime, the limiting energy depends on the magnetic field allowing to recover the Lorentz force in the semi-classical limit. Our solutions concentrate around global or local minima of a limiting energy that depends on the electric potential and the magnetic field. The results cover unbounded domains, fast-decaying electric potential and unbounded electromagnetic fields. The construction is variational and is based on an asymptotic analysis of solutions to a penalized problem in the spirit of M. del Pino and P. Felmer.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jonathan Di Cosmo, Jean Van Schaftingen. 2015-02-13. Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field. https://doi.org/10.1016/j.jde.2015.02.016

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Renormalized Lambert-W Cascade and Finite-Time Amplification and Blowup for reconstructed b Dynamics on T^3 for the 3D Navier Stokes Equations

This article extracts and consolidates the renormalized Lambert-$W$ branch-point cascade, its distinguished phase reduction, the exact characteristic invariant and finite-time amplification mechanism, and the extended reconstructed $b_i$ equation on $\mathbb T^3$. Repeated historical derivations are removed while the principal proofs and terminal reconstruction estimates are retained. The presentation separates exact finite-depth statements from coupled-depth asymptotics and records the hypotheses required for the extended PDE reconstruction. This paper further supports a recent paper \cite {moschandreou2026exploration} published by the corresponding author which claims that the Navier Stokes equations lose smoothness in finite time from initial smooth data.

math.AP↗
arXiv2026

Global harmonic analysis for $Φ^4_3$ on closed Riemannian manifolds

Following Parisi \& Wu's paradigm of stochastic quantization, we constructed in \cite{BDFT} a $Φ^4$ measure on an arbitrary closed, compact Riemannian manifold of dimension $3$ as an invariant measure of a singular stochastic partial differential equation. This solves a longstanding open problem in quantum fields on curved backgrounds. In the present work, we build all the harmonic and microlocal analysis tools that are needed in \cite{BDFT}. In particular, we extend the approach of Jagannath--Perkowski to the vectorial $Φ^4_3$ model by introducing a new Cole-Hopf transform involving random bundle maps.

math.AP↗
arXiv2026

Unconditional uniqueness for the derivative nonlinear Schrödinger equation by normal form approach

We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schrödinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the uniqueness in $C_tH^s_x$, $s>1/2$. To overcome logarithmic divergences at the $H^{1/2}$ regularity, we exploit the $B^{0+}_{\infty,1}$ control of solutions provided by a refined Strichartz estimate. Our NFR argument consists of two stages: we first use NFR finitely many times to derive an intermediate equation in which the main cubic nonlinearity is restricted to a certain type of frequency interaction; we then apply the infinite NFR scheme to the intermediate equation. Moreover, we modify the usual NFR argument relying on continuity in time of solutions so that the uniqueness in the class $L^\infty_tH^{1/2}_x$ can be obtained directly.

math.AP↗