Geodesic
On solution of the Cauchy problem for the Korteweg–de Vries equation with initial data the sum of a periodic and a rapidly decreasing function
Sbornik. Mathematics, Tome 63 (1989) no. 1, pp. 257-265 
N. E. Firsova. On solution of the Cauchy problem for the Korteweg–de Vries equation with initial data the sum of a periodic and a rapidly decreasing function. Sbornik. Mathematics, Tome 63 (1989) no. 1, pp. 257-265. http://geodesic.mathdoc.fr/item/SM_1989_63_1_a17/
@article{SM_1989_63_1_a17,
     author = {N. E. Firsova},
     title = {On solution of the {Cauchy} problem for the {Korteweg{\textendash}de~Vries} equation with initial data the sum of a~periodic and a~rapidly decreasing function},
     journal = {Sbornik. Mathematics},
     pages = {257--265},
     year = {1989},
     volume = {63},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1989_63_1_a17/}
}
TY  - JOUR
AU  - N. E. Firsova
TI  - On solution of the Cauchy problem for the Korteweg–de Vries equation with initial data the sum of a periodic and a rapidly decreasing function
JO  - Sbornik. Mathematics
PY  - 1989
SP  - 257
EP  - 265
VL  - 63
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1989_63_1_a17/
LA  - en
ID  - SM_1989_63_1_a17
ER  - 
%0 Journal Article
%A N. E. Firsova
%T On solution of the Cauchy problem for the Korteweg–de Vries equation with initial data the sum of a periodic and a rapidly decreasing function
%J Sbornik. Mathematics
%D 1989
%P 257-265
%V 63
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1989_63_1_a17/
%G en
%F SM_1989_63_1_a17

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

A scheme is presented for solving the Cauchy problem for the KdV equation with initial data a sum of a periodic function $p(x)$ and a rapidly decreasing function $q(x)$. The scattering theory constructed earlier by the author for the pair of operators $H_0=-d^2/dx^2+p(x)$ and $H=H_0+q(x)$ is used to solve this problem. Evolution formulas for the scattering data are found. The solution $p(x,t)$ of the KdV equation with a periodic initial condition obtained by V. A. Marchenko and S. P. Novikov is assumed known. Bibliography: 11 titles.

[1] Novikov S. P., “Periodicheskaya zadacha dlya uravneniya Kortevega-de Friza”, Funktsion. analiz i ego pril., 8:3 (1974), 54–66 | MR | Zbl

[2] Marchenko V. A., “Periodicheskaya zadacha dlya Kortevega-de Friza”, Matem. sb., 95(137) (1974), 331–356 | Zbl

[3] Marchenko V. A., Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR

[4] Firsova N. E., “Obratnaya zadacha rasseyaniya dlya vozmuschennogo operatora Khilla”, Matem. zametki, 18:6 (1975), 831 | MR | Zbl

[5] Firsova N. E., “Pryamaya i obratnaya zadacha rasseyaniya dlya odnomernogo vozmuschennogo operatora Khilla”, Matem. sb., 130(172) (1986), 349–385 | MR | Zbl

[6] Birman M. Sh., “O spektre singulyarnykh granichnykh zadach”, Matem. sb., 55(97) (1961), 125–174 | MR | Zbl

[7] Rofe-Beketov F. S., “Priznak konechnosti diskretnykh urovnei, vnosimykh v lakuny nepreryvnogo spektra vozmuscheniyami periodicheskogo potentsiala”, DAN SSSR, 156:3 (1964), 515–518 | MR | Zbl

[8] Firsova N. E., “Rimanova poverkhnost kvaziimpulsa i teoriya rasseyaniya dlya vozmuschennogo operatora Khilla”, Zap. nauchn. seminarov LOMI, 51 (1975), 183–196 | MR | Zbl

[9] Its A. R., Matveev V. B., “Ob odnom klasse resheniya uravneniya KdF”, Problemy matem. fiziki, 8, Izd-vo Leningr. un-t, 1976, 70–91 | MR

[10] Titchmarsh E. Ch., Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, T. 2, IL, M., 1961

[11] Matveev V. B., Abelian functions and solutions, Preprint No 373, 1976