Geodesic
A nonlinear eigenvalue problem
Teoretičeskaâ i matematičeskaâ fizika, Tome 157 (2008) no. 2, pp. 175-187 Cet article a éte moissonné depuis la source Math-Net.Ru

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We reduce the problem of the zeros of an entire function related to the Painlevé I equation to the eigenvalue problem for the Dirichlet problem for a fourth-order bilinear equation. In a simplified case, this problem reduces to the problem of the position of the zeros of theta functions. We then apply the developed general method to the inverse scattering problem.
Keywords: entire function, inverse scattering problem.
Mots-clés : Painlevé transcendent 
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M. Kh. Chankaev; A. B. Shabat. A nonlinear eigenvalue problem. Teoretičeskaâ i matematičeskaâ fizika, Tome 157 (2008) no. 2, pp. 175-187. http://geodesic.mathdoc.fr/item/TMF_2008_157_2_a1/

[1] A. B. Shabat, Inverse Problems, 8:2 (1992), 303–308 | DOI | MR | Zbl

[2] A. Degasperis, A. B. Shabat, TMF, 100:2 (1994), 230–247 | DOI | MR | Zbl

[3] V. E. Adler, “On the rational solutions of the Shabat equation”, Nonlinear Physics: Theory and Experiment, Proc. Int. Workshop, eds. E. Alfinito et al., World Scientific, River Edge, NJ, 1996, 3–10 | MR | Zbl

[4] Yunkang Liu, J. Differential Equations, 154:1 (1999), 1–41 | DOI | MR | Zbl

[5] E. Titchmarsh, Teoriya funktsii, Nauka, M., 1980 | MR | Zbl

[6] V. I. Gromak, I. Laine, S. Shimomura, Painlevé Differential Equations in the Complex Plane, de Gruyter Stud. Math., 28, Walter de Gruyter, Berlin, 2002 | DOI | MR | Zbl

[7] S. I. Pokhozhaev, DAN, 165 (1965), 36–39 | MR | Zbl

[8] R. N. Garifullin, Chastnoe soobschenie (2007)

[9] A. B. Shabat, Dinamika sploshnoi sredy, Sb. nauch. trudov, no. 5, AN SSSR, Institut gidrodinamiki, Novosibirsk, 1970, 130–145