Geodesic
Primitive solutions of the~Korteweg--de~Vries equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 3, pp. 382-392

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: integrable system, Korteweg–de Vries equation, primitive solution. 
@article{TMF_2020_202_3_a4,
     author = {S. A. Dyachenko and P. Nabelek and D. V. Zakharov and V. E. Zakharov},
     title = {Primitive solutions of {the~Korteweg--de~Vries} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {382--392},
     publisher = {mathdoc},
     volume = {202},
     number = {3},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2020_202_3_a4/}
}
TY  - JOUR
AU  - S. A. Dyachenko
AU  - P. Nabelek
AU  - D. V. Zakharov
AU  - V. E. Zakharov
TI  - Primitive solutions of the~Korteweg--de~Vries equation
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2020
SP  - 382
EP  - 392
VL  - 202
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TMF_2020_202_3_a4/
LA  - ru
ID  - TMF_2020_202_3_a4
ER  - 
%0 Journal Article
%A S. A. Dyachenko
%A P. Nabelek
%A D. V. Zakharov
%A V. E. Zakharov
%T Primitive solutions of the~Korteweg--de~Vries equation
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2020
%P 382-392
%V 202
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TMF_2020_202_3_a4/
%G ru
%F TMF_2020_202_3_a4
S. A. Dyachenko; P. Nabelek; D. V. Zakharov; V. E. Zakharov. Primitive solutions of the~Korteweg--de~Vries equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 3, pp. 382-392. http://geodesic.mathdoc.fr/item/TMF_2020_202_3_a4/