Canonical systems of partial differential equations
type: Item
instance of: scholarly article
title: Canonical systems of partial differential equations
zbMATH Open document ID: 1471.35126
DOI: 10.1007/978-3-030-72563-1_22
publication date: 2021-09-07T00:00:00Z
review text: The author discusses existence results for solutions of some specific elliptic PDE system consisting of two weakly coupled nonlinear Schrödinger equations with saturation effect describing the propogation of light in photonic lattices. It is given by \begin{align*} \Delta v &= \frac{Pv}{1+v^2+w^2} + \lambda v \quad\text{in }\Omega, \\ \Delta w &= \frac{Qw}{1+v^2+w^2} + \lambda w \quad\text{in }\Omega, \end{align*} where \(\lambda,P,Q\in\mathbb{R}\) and \(\Omega\subset\mathbb{R}^n\). In his main results (Theorems~1--6) the author identifies various parameter regimes where the existence of nontrivial solutions can be proved using variational methods such as global minimization (Theorem 1) and well-known Linking Theorems (Theorems~2--6). Reviewer's remark: In the case of ground state solutions (i.e., nontrivial solutions of least energy) more powerful results than the given ones are possible by a comparison of the two scalar least energy levels and the least energy level of the system. If the latter is smaller than the former, the ground state is known to be vectorial, i.e., one has \(v\neq 0\) and \(w\neq 0\). Such arguments have been presented in similar contexts in [Zbl 1286.35104; Zbl 1130.34014; Zbl 1257.35171; Zbl 1104.35053; Zbl 1135.35072; Zbl 1334.35319; Zbl 1312.35082]. Moreover, one may add that vectorial solutions can be obtained by different methods such as bifurcation theory. For the entire collection see [Zbl 1471.34005].
reviewed by: Rainer Mandel
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 35J47
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 7392157
author: Martin Schechter
MaRDI profile type: Publication
full work available at URL: 978-3-030-72563-1 22
OpenAlex ID: W3194477017
cites work: Existence of steady-state solutions in a nonlinear photonic lattice model
cites work: The mathematics of photonic crystals
cites work: On the steady-state solutions of a nonlinear photonic lattice model
cites work: Periodic nonlinear Schrödinger equation with application to photonic crystals
cites work: Q4223582
cites work: Q3158585
cites work: The use of Cerami sequences in critical point theory
cites work: Minimax Systems and Critical Point Theory
cites work: Steady state solutions for Schrödinger equations governing nonlinear optics
cites work: Photonic lattices
cites work: Critical Point Theory
cites work: Schrödinger equations in nonlinear optics
cites work: Q2734533
cites work: Steady state solutions for nonlinear Schrödinger equation arising in optics
cites work: Dipole and Quadrupole Solitons in Optically Induced Two‐Dimensional Photonic Lattices: Theory and Experiment
description: scientific article; zbMATH DE number 7392157