Geodesic
Existence of a family of soliton-like solutions for the Kawahara equation
Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 42-50 
A. T. Il'ichev. Existence of a family of soliton-like solutions for the Kawahara equation. Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 42-50. http://geodesic.mathdoc.fr/item/MZM_1992_52_1_a6/
@article{MZM_1992_52_1_a6,
     author = {A. T. Il'ichev},
     title = {Existence of a family of soliton-like solutions for the {Kawahara} equation},
     journal = {Matemati\v{c}eskie zametki},
     pages = {42--50},
     year = {1992},
     volume = {52},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1992_52_1_a6/}
}
TY  - JOUR
AU  - A. T. Il'ichev
TI  - Existence of a family of soliton-like solutions for the Kawahara equation
JO  - Matematičeskie zametki
PY  - 1992
SP  - 42
EP  - 50
VL  - 52
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_1992_52_1_a6/
LA  - ru
ID  - MZM_1992_52_1_a6
ER  - 
%0 Journal Article
%A A. T. Il'ichev
%T Existence of a family of soliton-like solutions for the Kawahara equation
%J Matematičeskie zametki
%D 1992
%P 42-50
%V 52
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_1992_52_1_a6/
%G ru
%F MZM_1992_52_1_a6

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

Existence is proved for a family of soliton-like solutions for the nonlinear evolution equation $\mathbf{u}_t+\mathbf{uu}_x+\mathbf{u}_{xxx}-\mathbf{u}_{xxxxx}=0$. The problem is reduced to investigating the fixed points of the operator $$ (Au)(x)=\int_{-\infty}^{\infty}k(x-y)u^2(y)\,dy, \quad \int_{-\infty}^{\infty}k(x)=1, $$ whose action is considered in a cone of Frechet functions that are continuous on the real axis.