Geodesic
Compactness of the set of multisoliton solutions of the nonlinear Schrödinger equation
Sbornik. Mathematics, Tome 75 (1993) no. 2, pp. 429-443 Cet article a éte moissonné depuis la source Math-Net.Ru

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Multisoliton solutions $\psi(x,t)$ of the nonlinear Schrödinger equation are considered which satisfy the condition of finite density: $$ \lim_{x\to\pm\infty}\psi(x,t)=\frac12\omega e^{i\psi_\pm}. $$ It is proved that all these solutions satisfy the inequalities $$ \sup_{\substack{-\infty<x<\infty\\-\infty<t<\infty}}\biggl|\frac{\partial^m} {\partial t^m}\frac{\partial^n}{\partial x^n}\psi(x,\,t)\biggr|\leqslant\frac14 (2\omega)^{1+n+2m}(n+2m)! $$ ($m,n=0,1,2,\dots$), which implies solvability of the Cauchy problem for the nonlinear Schrödinger equation with an initial function $\psi(x,0)$ belonging to the closure of the set of nonreflecting potentials. 
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     author = {D. Sh. Lundina and V. A. Marchenko},
     title = {Compactness of the set of multisoliton solutions of the nonlinear {Schr\"odinger} equation},
     journal = {Sbornik. Mathematics},
     pages = {429--443},
     year = {1993},
     volume = {75},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1993_75_2_a6/}
}
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D. Sh. Lundina; V. A. Marchenko. Compactness of the set of multisoliton solutions of the nonlinear Schrödinger equation. Sbornik. Mathematics, Tome 75 (1993) no. 2, pp. 429-443. http://geodesic.mathdoc.fr/item/SM_1993_75_2_a6/

[1] Zakharov V. E., Shabat A. B., “Tochnaya teoriya dvumernoi samofokusirovki i odnomernoi avtomodulyatsii voln v nelineinykh sredakh”, ZhETF, 61:1 (1971), 118–134 | MR

[2] Zakharov V. E., Shabat A. B., “O vzaimodeistvii solitonov v ustoichivoi srede”, ZhETF, 64:5 (1973), 1627–1639

[3] Takhtadzhyan L. A., Faddeev L. D., Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | Zbl

[4] Marchenko V. A., “Zadacha Koshi dlya uravneniya Kortevega - de Friza s neubyvayuschimi nachalnymi dannymi”, Integriruemost i kinematicheskie uravneniya dlya solitonov, Naukova dumka, Kiev, 1990, 168–211 | MR

[5] Lundina D. Sh., “Kompaktnost mnozhestva bezotrazhatelnykh potentsialov”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, vyp. 44, Kharkov, 1985, 57–67