Primitive solutions of the Korteweg–de Vries equation

S. A. Dyachenko, P. Nabelek, D. V. Zakharov, V. E. Zakharov

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions. These solutions are constructed as limits of rapidly vanishing solutions of the Korteweg-de Vries equation as the number of solitons tends to infinity. A primitive solution is determined nonuniquely by a pair of positive functions on an interval on the imaginary axis and a function on the real axis determining the reflection coefficient. We show that elliptic one-gap solutions and, more generally, periodic finite-gap solutions are special cases of reflectionless primitive solutions.

Original languageEnglish
Pages (from-to)334-343
Number of pages10
JournalTheoretical and Mathematical Physics
Issue number3
StatePublished - Mar 1 2020


  • Korteweg-de Vries equation
  • integrable system
  • primitive solution


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