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.
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The research of S. A. Dyachenko and D. V. Zakharov was supported by the National Science Foundation (Grant No. DMS-1716822)
The research of V. E. Zakharov was supported by the National Science Foundation (Grant No. DMS-1715323). The results in Secs. 3-5 were obtained with support of a grant from the Russian Science Foundation (Project No. 19-72-30028)
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 202, No. 3, pp. 382–392, March, 2020.
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Dyachenko, S.A., Nabelek, P., Zakharov, D.V. et al. Primitive solutions of the Korteweg–de Vries equation. Theor Math Phys 202, 334–343 (2020). https://doi.org/10.1134/S0040577920030058
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DOI: https://doi.org/10.1134/S0040577920030058