Geodesic
Birth of step-like contrast structures connected with a cusp catastrophe
Sbornik. Mathematics, Tome 195 (2004) no. 12, pp. 1727-1746

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A special solution of the ordinary differential equation $u''_{xx}=u^3-tu-x$ is considered which relates to solutions of a broad spectrum of partial differential equations with a small parameter. The function $u(x,t)$ is the dominant term of asymptotic expressions with respect to the small parameter for these solutions near cusp points of the limiting solution. The existence of this special function $u(x,t)$ is proved; its uniform asymptotics at infinity are constructed and substantiated. 
@article{SM_2004_195_12_a1,
     author = {A. M. Il'in and B. I. Suleimanov},
     title = {Birth of step-like contrast structures connected with a cusp catastrophe},
     journal = {Sbornik. Mathematics},
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     volume = {195},
     number = {12},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2004_195_12_a1/}
}
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A. M. Il'in; B. I. Suleimanov. Birth of step-like contrast structures connected with a cusp catastrophe. Sbornik. Mathematics, Tome 195 (2004) no. 12, pp. 1727-1746. http://geodesic.mathdoc.fr/item/SM_2004_195_12_a1/