Russian Journal of Mathematical Physics

, Volume 13, Issue 1, pp 4–12 | Cite as

On the variational representation of solutions to some quasilinear equations and systems of hyperbolic type on the basis of potential theory

  • A. I. Aptekarev
  • Yu. G. Rykov


We demonstrate a method that permits to obtain generalized solutions for some quasilinear equations and systems of hyperbolic type. The corresponding variational principle is constructed using the theory of equilibrium of a potential in an external field.


Generalize Solution Variational Principle External Field Potential Theory Variational Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. D. Lax, “Hyperbolic Systems of Conservation Laws, II,” Comm. Pure Appl. Math. 10(4), 537–566 (1957).zbMATHMathSciNetGoogle Scholar
  2. 2.
    O. A. Oleinik, “The Cauchy Problem for Nonlinear Differential Equations of First Order with Discontinuous Initial Conditions,” Trudy Mosk. Mat. Obshch. 5, 433–454 (1956).zbMATHMathSciNetGoogle Scholar
  3. 3.
    W. E, Yu. G. Rykov and Ya. G. Sinai, “Generalized Variational Principles, Global Weak Solutions and Behavior with Random Initial Data for Systems of Conservation Laws Arising in Adhesion Particle Dynamics,” Comm. Math. Phys. 177, 349–380 (1996).ADSMathSciNetGoogle Scholar
  4. 4.
    Ph. Le Floch, “Shock Waves for Nonlinear Hyperbolic Systems in Nonconservative Form,” Preprint No. 593 (Institute Math. Appl., Minneapolis, 1989).Google Scholar
  5. 5.
    P. Deift and K. T.-R. McLaughlin, A Continuum Limit of the Toda Lattice, Memoirs Amer. Math. Soc. 624 (Providence, 1998).Google Scholar
  6. 6.
    A. I. Aptekarev and W. Van Assche, “Asymptotics of Discrete Orthogonal Polynomials and the Continuum Limit of the Toda Lattice,” J. Phys. A: Mathematics and General 34, 10627–10637 (2001).ADSGoogle Scholar
  7. 7.
    A. A. Gonchar and E. A. Rakhmanov, “Equilibrium Measure and the Distribution of Zeros of Extremal Polynomials,” Mat. Sb. 125(167), 117–127 (1984).MathSciNetGoogle Scholar
  8. 8.
    E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Grundlehren Math. Wiss. 316 (Springer, Berlin, 1997).Google Scholar
  9. 9.
    V. S. Buyarov and E. A. Rakhmanov, “On Families of Equilibrium Measures in an External Field on the Real Line,” Mat. Sb. 190(5), 11–22 (1999) [Russian Acad. Sci. Sb. Math. 190, 791–802 (1999)].MathSciNetGoogle Scholar
  10. 10.
    H. N. Mhaskar and E. B. Saff, “Where Does the Sup Norm of a Weighted Polynomial Live?” Constr. Approx. 1, 71–91 (1985).CrossRefMathSciNetGoogle Scholar

Copyright information

© MAIK “Nauka/Interperiodica” 2006

Authors and Affiliations

  • A. I. Aptekarev
    • 1
  • Yu. G. Rykov
    • 1
  1. 1.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia

Personalised recommendations