Primitive potentials and bounded solutions of the KdV equation

S. Dyachenko, D. Zakharov, V. Zakharov

Research output: Contribution to journalArticlepeer-review

23 Scopus citations


We construct a broad class of bounded potentials of the one-dimensional Schrödinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hölder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.

Original languageEnglish (US)
Pages (from-to)148-156
Number of pages9
JournalPhysica D: Nonlinear Phenomena
StatePublished - Oct 15 2016


  • Integrability
  • Schrödinger operator
  • Solitonic gas

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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