Geodesic
Formation of wavy nanostructures on the surface of flat substrates by ion bombardment
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 5, pp. 930-945

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A popular mathematical model for the formation of an inhomogeneous topography on the surface of a plate (flat substrate) during ion bombardment was considered. The model is described by the Bradley–Harper equation, which is frequently referred to as the generalized Kuramoto–Sivashinsky equation. It was shown that inhomogeneous topography (nanostructures in the modern terminology) can arise when the stability of the plane incident wavefront changes. The problem was solved using the theory of dynamical systems with an infinite-dimensional phase space, in conjunction with the integral manifold method and Poincaré–Dulac normal forms. A normal form was constructed using a modified Krylov–Bogolyubov algorithm that applies to nonlinear evolutionary boundary value problems. As a result, asymptotic formulas for solutions of the given nonlinear boundary value problem were derived. 
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     author = {A. N. Kulikov and D. A. Kulikov},
     title = {Formation of wavy nanostructures on the surface of flat substrates by ion bombardment},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {930--945},
     publisher = {mathdoc},
     volume = {52},
     number = {5},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_5_a14/}
}
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A. N. Kulikov; D. A. Kulikov. Formation of wavy nanostructures on the surface of flat substrates by ion bombardment. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 5, pp. 930-945. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_5_a14/