### Abstract

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 language | English (US) |
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Pages (from-to) | 334-343 |

Number of pages | 10 |

Journal | Theoretical and Mathematical Physics(Russian Federation) |

Volume | 202 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2020 |

### Keywords

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

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Dyachenko, S. A., Nabelek, P., Zakharov, D. V., & Zakharov, V. E. (2020). Primitive solutions of the Korteweg–de Vries equation.

*Theoretical and Mathematical Physics(Russian Federation)*,*202*(3), 334-343. https://doi.org/10.1134/S0040577920030058