Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.24261

Global well-posedness and orbital stability of the Gross--Pitaevskii branch in a coupled defocusing Schrödinger system

Michał Kowalczyk·Jarosław Mederski·Isidora Miranda
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider a one-dimensional coupled defocusing Schrödinger system with a nonzero background in its first component and finite mass in its second component. We prove global well-posedness in the full energy space, together with conservation of energy, bright mass, and total untwisted momentum. For all nonnegative coupling coefficients, every subsonic Gross--Pitaevskii soliton with vanishing second component is orbitally stable against perturbations of both components, including the zero-speed black soliton. At nonzero speed the proof uses a fixed energy--momentum functional augmented by the conserved bright mass. At zero speed it uses weighted coercivity and a local real lift of the momentum. We also identify the sharp threshold for positivity of the unaugmented bright energy and explain why this threshold is not a necessary condition for orbital stability.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michał Kowalczyk, Jarosław Mederski, Isidora Miranda. 2026-10-16. Global well-posedness and orbital stability of the Gross--Pitaevskii branch in a coupled defocusing Schrödinger system. https://arxiv.org/abs/2610.24261

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

KEEP EXPLORING

Related papers

arXiv2026

Well-Posedness of Fully Nonlinear PDEs on Star-Shaped Networks with Nonlinear Kirchhoff Conditions

The aim of this paper is to establish the well-posedness of fully nonlinear partial differential equations on star-shaped networks subject to nonlinear Kirchhoff conditions at the junction. The branchwise operators may be discontinuous across the junction, and uniform ellipticity is required only in a neighborhood of the vertex. Our approach relies on the construction of auxiliary test functions solving suitable nonlinear ordinary differential equations, whose structure allows us to control the viscosity inequalities at the junction. We first show that generalized Kirchhoff viscosity solutions in the sense of Lions-Souganidis satisfy the strong Kirchhoff condition. Combining these auxiliary functions with an affine perturbation of supersolutions and precise junction gradient estimates, we establish a comparison principle for bounded semicontinuous viscosity sub- and supersolutions. Existence and uniqueness are then addressed through Perron's method and the comparison principle. Finally, we discuss first-order junction problems and a possible transmission condition arising from vanishing viscosity.

math.AP↗
arXiv2026

Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian

We prove existence and regularity results for the following elliptic system: \[ \begin{cases} -\textbf{div}(|D\boldsymbol{u}|^{p-2}D\boldsymbol{u})=\boldsymbol{f}(x,\boldsymbol{u}) & \text{in } Ω\\ \boldsymbol{u}=0 & \text{on } \partialΩ, \end{cases} \] where $\boldsymbol{u}=(u^1,\dots,u^m)$, $p>1$, and $Ω\subset\mathbb{R}^N$ is a bounded domain. We also consider the special case \[\boldsymbol{f}(x,\boldsymbol{u})=λ|\boldsymbol{u}|^{p-2}\boldsymbol{u}+|\boldsymbol{u}|^{q-2}\boldsymbol{u},\] and we prove a classification result. In particular, we show that any least energy solution is of the form $(c^1ω,\dots,c^mω)$, where $\boldsymbol{c}=(c^1,\dots,c^m)\in S^{m-1}$ (the $(m-1)$-sphere in $\mathbb R^m$) and $ω$ is a positive solution of the corresponding scalar equation.

math.AP↗
arXiv2026

Feedback Stabilization and Tracking for Heat Equations Using Thermo-Plasmonic Nanoparticles as Actuators

We propose a feedback strategy to track prescribed heat profiles using plasmonic nanoparticles as actuators. Starting from a thermo--plasmonic Maxwell--heat model, we use a time-domain discrete effective description in which the generated heat is approximated by a superposition of heat kernels centered at particle locations with amplitudes governed by a coupled Volterra system. We recast this dynamics as a heat equation on a bounded domain with finitely many point actuators and design a tracking feedback based on pointwise evaluations of $\mathcal A^{-1}y$, where $\mathcal A=I-A_0$ and $A_0$ is the Neumann diffusion operator. Working in the natural $V'$ setting with $V=D(\mathcal A)$, we prove exponential stabilization of the tracking error via distribution-actuator theory. For non-equilibrium reference profiles, we add a constant feedforward term and a low-mode fixed-point pre-compensation on $X_N$, ensuring exact steady matching on $X_N$ and an explicit bound on the residual tail mismatch.

math.AP↗