Geodesic
On the absence of global periodic solutions of a  Schrödinger-type nonlinear evolution equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 208 (2021) no. 1, pp. 69-73 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the problem of the absence of global periodic solutions of a nonlinear Schrödinger-type evolution equation with a damped linear term. We prove that in the case where the damping factor is negative, the problem has no global periodic solutions for any initial data, and in the case of a negative damping factor, this is also true for large enough values of the initial data.
Keywords: nonlinear evolution equation, Schrödinger equation, periodic solution, absence of periodic global solutions.
Mots-clés : global solution 
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Sh. M. Nasibov. On the absence of global periodic solutions of a  Schrödinger-type nonlinear evolution equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 208 (2021) no. 1, pp. 69-73. http://geodesic.mathdoc.fr/item/TMF_2021_208_1_a4/

[1] J. J. Rasmussen, K. Rypdal, “Blow-up in nonlinear Schrodinger equations-I. A general review”, Phys. Scr., 33:6 (1986), 481–497 ; “Blow-up in nonlinear Schrodinger equations-II. Similarity structure of the blow-up singularity”, 498–504 | DOI | MR | MR

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

[3] V. N. Lugovoi, A. M. Prokhorov, “Teoriya rasprostraneniya moschnogo lazernogo izucheniya v lineinoi srede”, UFN, 11:10 (1973), 203–247 | DOI | DOI

[4] J. Bourgain, “Exponential sums and nonlinear Schrodinger equations”, Geom. Funct. Anal., 3:2 (1993), 157–178 | DOI | MR

[5] A. B. Shabat, “O zadache Koshi dlya uravneniya Ginzburga–Landau”, Dinamika sploshnoi sredy, Vyp. 1, Izd-vo In-ta gidrodinamiki SO RAN, Novosibirsk, 1969, 180–194

[6] Sh. M. Nasibov, “O razrushenii reshenii smeshannoi zadachi dlya nelineinogo evolyutsionnogo uravneniya Ginzburga–Landau–Shredingera”, Differents. uravneniya, 39:8 (2003), 1087–1091 | DOI | MR

[7] Sh. M. Nasibov, “On some sufficient conditions for the blow-up solutions of the nonlinear Ginzburg–Landau–Schrödinger evolution equation”, J. Appl. Math., 2004:1 (2004), 23–35 | DOI | MR

[8] Sh. M. Nasibov, “Ob otsutstvii globalnykh reshenii smeshannoi zadachi dlya nelineinogo evolyutsionnogo uravneniya tipa Shredingera”, TMF, 195:2 (2018), 190–196 | DOI | DOI | MR

[9] Sh. M. Nasibov, “Nelineinoe evolyutsionnoe uravnenie Shredingera v dvumernoi oblasti”, TMF, 201:1 (2019), 118–125 | DOI | DOI | MR

[10] Sh. M. Nasibov, “Ob odnom nelineinom uravnenii Shredingera s dissipativnym chlenom”, Dokl. AN SSSR, 304:2 (1989), 285–289 | MR | Zbl

[11] Sh. M. Nasibov, “Ob ustoichivosti, razrushenii, zatukhanii i samokanalizatsii reshenii odnogo nelineinogo uravneniya Shredingera”, Dokl. AN SSSR, 285:4 (1985), 807–811 | MR | Zbl

[12] Sh. M. Nasibov, “Ob optimalnykh konstantakh v nekotorykh neravenstvakh Soboleva i ikh prilozhenii k nelineinomu uravneniyu Shredingera”, Dokl. AN SSSR, 307:3 (1989), 538–542 | MR | Zbl

[13] O. I. Kudryashov, “Ob osobennostyakh reshenii nelineinykh uravnenii tipa Ginzburga–\allowbreakLandau”, Sib. matem. zhurnal, 16:4 (1975), 866–868 | DOI | MR | Zbl

[14] M. Weinstein, “On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations”, Commun. Partial Differential Equations, 11:5 (1986), 545–565 | DOI | MR

[15] H. Nawa, “Asymptotic and limiting profiles of blow up solutions of the nonlinear Schrödinger equations with critical power”, Commun. Pure Appl. Math., 52:2 (1999), 193–270 | 3.0.CO;2-3 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[16] Sh. M. Nasibov, “Ob otsutstvii globalnykh periodicheskikh reshenii nelineinogo evolyutsionnogo uravneniya tipa Shredingera”, Dokl. RAN. Matematika, informatika, protsessy upravleniya, 494:1 (2020), 53–55 | DOI