Hamiltonian structure of ``Thirring \(\times\) Liouville'' model. Singular solutions
type: Item
instance of: scholarly article
title: Hamiltonian structure of ``Thirring \(\times\) Liouville'' model. Singular solutions
zbMATH Open document ID: 0649.35078
DOI: 10.1007/BF01017091
author: S. V. Talalov
published in: Theoretical and Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: In two-dimensional space-time, interacting scalar and spinor fields possessing definite singularities in the (t,x) plane are cn (- \(\infty,0]\), \(g(-1)=0\), g(u)\(\leq 0\) on (-\(\infty,-1]\) 0\(\leq \ell \leq 1\), \(\phi \in H^ 1_ 0(0,1)\), \(\phi (\ell)=0\), (\(\ell - x)\phi (x)\geq 0,0\leq x\leq 1.\) This problem of the Stefan type with sources was studied by the authors in previous papers, under non-homogeneous Dirichlet data, where the free boundary s(t) never touches \(x=0,1\). In the present case it is possible that one phase disappear in finite time. The authors study global existence, uniqueness, regularity and asymptotic properties of solutions. Attention is given to the possibility that \(s(T^*)\) be either 0 or 1 for certain \(T^*:\) the authors show that in that case s(t) will be identically 0 or 1 for \(t\geq T^*\), and therefore the problem becomes an initial-boundary value problem for \(t\geq T^*\). Comparison theorems are given, and results on the dependence of solutions on the initial data \(\phi\),\(\ell\) are proven. These results are used to study the \(\omega\)-limit set corresponding to the solution orbit of the problem: every element in this set satisfies the associated stationary problem. Stability and bifurcation are also studied.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35G20
Mathematics Subject Classification ID: 35A20
zbMATH DE Number: 4059984
zbMATH Keywords: essentially nonlinear evolution equations
zbMATH Keywords: two-dimensional space-time
zbMATH Keywords: interacting scalar and spinor fields
zbMATH Keywords: definite singularities
zbMATH Keywords: Stefan type
zbMATH Keywords: Dirichlet data
zbMATH Keywords: global existence
zbMATH Keywords: uniqueness
zbMATH Keywords: regularity
zbMATH Keywords: asymptotic properties
zbMATH Keywords: Comparison theorems
zbMATH Keywords: dependence of solutions on the initial data
zbMATH Keywords: \(\omega \)-limit set
zbMATH Keywords: stationary problem
zbMATH Keywords: Stability
zbMATH Keywords: bifurcation
reviewed by: Julio E. Bouillet
MaRDI profile type: Publication
cites work: Liouville field theory: IST and Poisson bracket structure
cites work: A simple model of the integrable Hamiltonian equation
cites work: Q3769640
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cites work: Q3732103
description: scientific article; zbMATH DE number 4059984