B\"{a}cklund transformations for the one-dimensional Schr\"{o}dinger equation

M. V. Neshchadim

Sibirskij žurnal industrialʹnoj matematiki, Tome 24 (2021) no. 2, pp. 116-125.

Résumé

We study the system of equations which bases on the one-dimensional Schrödinger equation and connects the potential, amplitude, and phase functions. Using the methods of compatibility theory of systems of partial differential equations, we obtain the completely integrable systems that connect only two functions of the above three. As a corollary, we construct some exact solutions of the Schrödinger equation.

Détail

Keywords: Schrödinger equation, Bäcklund transformation, compatibility condition.

Bibliographie

[1] L. V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, N.Y., 1982 | MR | Zbl

[2] N. Kh. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel Publ. Company, Dordrecht–Boston–Lancaster, 1985 | MR | MR | Zbl

[3] P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verl, N.Y., 1986 | MR | Zbl

[4] A. M. Vinogradov, Symmetries and conservation laws for differential equations of mathematical physics, Faktorial, M., 1997 (in Russian)

[5] A. F. Sidorov, V. P. Shapeev, N. N. Yanenko, Method of differential constraints and its applications in gas dynamics, Nauka, Novosibirsk, 1984 (in Russian)

[6] N. Kh. Ibragimov, A. B. Shabat, “Evolutionary equations with a nontrivial Lie-Bäcklund group”, Function. Anal. and Its Appl., 14:1 (1980), 25–36 | MR | Zbl

[7] N. Kh. Ibragimov, A. B. Shabat, “Infinite Lie-Bäcklund Algebras”, Function. Anal. and Its Appl., 14:4 (1980), 79–80 (in Russian) | MR | Zbl

[8] A. M. Vinogradov, I. S. Krasilshchik, V. V. Lychagin, Introduction to the geometry of nonlinear differential equations, Nauka, M., 1983 (in Russian)

[9] A. M. Vinogradov, Cohomological Analysis of Partial Differential Equations and Secondary Calculus, Amer. Math. Soc., Providence, 2021 | MR

[10] R. M. Miura, Backlund Transformations, Lect. Notes Math., 515, Springer-Verl., Heidelberg, 1976 | DOI | MR | Zbl

[11] V. V. Zharinov, “On Bäcklund Correspondences”, Math. USSR-Sb, 64:1 (1989), 277–293 | DOI | MR | Zbl

[12] V. V. Zharinov, “Bäcklund correspondences for evolution equations in a multidimensional space”, Theor. Math. Phys., 147:1 (2006), 449–459 | DOI | MR | Zbl

[13] Yu. E. Anikonov, M. V. Neshchadim, “Generalized Cole-Hopf transformation”, J. Appl. Ind. Math., 21:3 (2018), 409–416 | DOI | MR | Zbl

[14] Yu. E. Anikonov, M. V. Neshchadim, “The method of differential constraints and nonlinear inverse problems”, J. Appl. Ind. Math., 9:3 (2015), 317–327 | DOI | MR | Zbl

[15] M. J. Ablovits, Kh. Sigur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1987 | MR

[16] O. V. Kaptsov, Integration methods for partial differential equations, Fizmatlit, M., 2009 (in Russian)

[17] T. Miva, M. Dzhimbo, E. Date, Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, Jwanami Shoten, Tokyo, 1993

[18] D. I. Blokhintsev, Fundamentals of quantum mechanics, Nauka, M., 1976 (in Russian)

[19] S. P. Finikov, Cartan's method of exterior forms, Gostekhizdat, M.–L,, 1948 (in Russian)

[20] Zh. Pommare, Systems of partial differential equations and Lie pseudogroups, 1983, Mir, M. (in Russian)

[21] M. V. Neshchadim, A. P. Chupakhin, “Partially invariant solutions to the cubic Schrödinger equation”, Vestnik Udmurt. Univ., 2008, no. 3, 35–41 (in Russian)