The Mhaskar–Saff Variational Principle and Location of the Shocks of Certain Hyperbolic Equations

We discuss an application of the Mhaskar–Saff functional to the problem of location of the hyperbolic shocks in the context of completely integrable approximations to nonlinear hyperbolic Partial Differential Equations (PDEs) which exhibit shock formation. Different families of completely integrable...

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Bibliographic Details
Published inModern Trends in Constructive Function Theory Vol. 661; pp. 167 - 186
Main Author Aptekarev, A. I.
Format Book Chapter
LanguageEnglish
Published Providence, Rhode Island American Mathematical Society 31.03.2016
SeriesContemporary Mathematics
Subjects
difference operators
hyperbolic PDEs
recurrence relations Hermite-Padé approximants
spectral measures
Logarithmic potential
orthogonal polynomials
integrable systems
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ISBN1470425343
9781470425340
ISSN0271-4132
1098-3627
DOI10.1090/conm/661/13281

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Summary:We discuss an application of the Mhaskar–Saff functional to the problem of location of the hyperbolic shocks in the context of completely integrable approximations to nonlinear hyperbolic Partial Differential Equations (PDEs) which exhibit shock formation. Different families of completely integrable systems admit interpretation as semidiscrete approximations to hyperbolic PDEs, the Toda lattice being a famous example. For the limiting PDEs an inverse spectral problem method based on the logarithmic potential with external field theory is considered. Special attention is focused on multidimensional (in space variables) generalizations.
ISBN:1470425343
9781470425340
ISSN:0271-4132
1098-3627
DOI:10.1090/conm/661/13281