The solvability of the initial-boundary value problems for a nonlinear Schrödinger equation with a special gradient term
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 2, pp. 214-232 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, the initial-boundary value problems for the two-dimensional nonlinear Schrödinger equation with a special gradient term with purely imaginary coefficients in the nonlinear part, when the coefficients of the equation are measurable bounded functions, are considered. The existence and uniqueness of solutions of the first and second initial-boundary value problems is proved almost everywhere.
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G. Yagub; N. S. Ibrahimov; M. Zengin. The solvability of the initial-boundary value problems for a nonlinear Schrödinger equation with a special gradient term. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 2, pp. 214-232. http://geodesic.mathdoc.fr/item/JMAG_2018_14_2_a6/

[1] G. D. Akbaba, The Optimal Control Problem with the Lions Functional for the Schrödinger Equation Including Virtual Coefficient Gradient, Master's thesis, Kars, Turkey, 2011 (Turkish)

[2] L. Baudouin, O. Kavian, J. P. Puel, “Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control”, J. Differential Equations, 216 (2005), 188–222 | DOI | MR | Zbl

[3] A. G. Butkovskiy, Y. I. Samojlenko, Control of Quantum-Mechanical Processes and Systems, Mathematics and its Applications (Soviet Series), 56, Kluwer Academic Publishers Group, Dordrecht, 1990 | DOI | MR | Zbl

[4] N. S. Ibragimov, “The solvability of the initial-boundary value problems for the nonlinear stationary equation of quasi-optics with purely imaginary coefficient in the nonlinear part”, News of Baku State University, Ser. Physics and Math. Sciences, 2010, No. 3, 72–84

[5] A. D. Iskenderov, G. Y. Yagubov, “A variational method for solving an inverse problem of determining the quantum mechanical potential”, Dokl. Akad. Nauk SSSR, 303 (1988), 1044–1048 (Russian); Engl. transl.: Soviet Math. Dokl. | MR

[6] Automat. Remote Control, 50:12-1 (1989), 1631–1641 | MR | Zbl

[7] A. Iskenderov, G. Yagubov, “Optimal control of the unbounded potential in the multidimensional nonlinear nonstationary Schrödinger equation”, Bulletin of Lankaran State University, Ser. Natural Sciences, 2007, 3–56

[8] A. D. Iskenderov, G. Y. Yagubov, M. A. Musayeva, Identification of the Quantum Potentials, Chashyoglu, Baku, 2012 (Azerbaijani)

[9] M. Jahanshahi, S. Ashrafi, N. Aliev, “Boundary layer problem for the system of the first order ordinary differential equations with constant coefficients by general nonlocal boundary conditions”, Adv. Math. Models Appl., 2 (2017), 107–116

[10] O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics, Nauka, M., 1973 (Russian) | MR

[11] Translations of Mathematical Monographs, 23, Amer. Math. Soc., Providence, R.I., 1968 | DOI | MR | Zbl

[12] J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, v. I, Die Grundlehren der mathematischen Wissenschaften, 181, Springer-Verlag, New York–Heidelberg, 1972 | MR | Zbl

[13] L. S. Pontryagin, Ordinary Differential Equations, Nauka, M., 1982 | MR | Zbl

[14] F. P. Vasilyev, Numerical Methods for Solving of the Extremal Problems, Nauka, M., 1980 | MR

[15] M. A. Vorontsov, V. I. Schmalhausen, The Principles of Adaptive Optics, Nauka, M., 1985 (Russian) | MR

[16] G. Y. Yagubov, M. A. Musayeva, “On an identification problem for nonlinear Schrödinger equation”, Differ. Uravn., 33 (1997), 1691–1698 | MR | Zbl

[17] G. Yagubov, F. Toyğolu, M. Subaşı, “An optimal control problem for two-dimensional Schrödinger equation”, Appl. Math. Comput., 218 (2012), 6177–6187 | MR | Zbl

[18] K. Yajima, G. Zhang, “Smoothing property for Schrödinger equations with potential superquadratic at infinity”, Comm. Math. Phys., 221 (2001), 573–590 | DOI | MR | Zbl

[19] V. M. Zhuravlev, Nonlinear Waves in Multicomponent Systems Dispersion and Diffusion, Ulyanovsk State University, Ulyanovsk, 2001 (Russian)