Geodesic
Existence of a family of soliton-like solutions for the Kawahara equation
Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 42-50

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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. 
@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},
     publisher = {mathdoc},
     volume = {52},
     number = {1},
     year = {1992},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1992_52_1_a6/}
}
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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/