Third-Order Nonlinear Dispersive Equations: Shocks, Rarefaction, and Blowup Waves

Victor A Galaktionov, S I Pohozaev

Research output: Contribution to journalArticlepeer-review

34 Citations (Scopus)

Abstract

Shock waves and blowup arising in third-order nonlinear dispersive equations are studied. The underlying model is the equation u(t) = (uu(x))(xx) in R x R+. It is shown that two basic Riemann problems for Eq. (0.1) with the initial data S--/+ (x) = -/+ sgnx exhibit a shock wave (u(x, t) = S-(x)) and a smooth rarefaction wave (for S+), respectively. Various blowing-up and global similarity solutions to Eq. (0.1) are constructed that demonstrate the fine structure of shock and rarefaction waves. A technique based on eigenfunctions and the nonlinear capacity is developed to prove the blowup of solutions. The analysis of Eq. (0.1) resembles the entropy theory of scalar conservation laws of the form u(t) + uu(x) = 0, which was developed by O.A. Oleinik and S. N. Kruzhkov (for equations in x is an element of R-N) in the 1950s-1960s.
Original languageEnglish
Pages (from-to)1784-1810
Number of pages27
JournalComputational Mathematics and Mathematical Physics
Volume48
Issue number10
DOIs
Publication statusPublished - Oct 2008

Keywords

  • Riemann's problem
  • shock waves
  • nonlinear dispersive
  • rarefaction and blowup waves
  • entropy theory of scalar conservation laws
  • equations
  • general theory of partial differential equations

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