Generalized solutions of the second equation of the Rickati hierarchy

E. V. Kuzmina

Problemy fiziki, matematiki i tehniki, no. 2 (2022), pp. 68-75.

Résumé

A nonlinear differential equation of the second order of the Riccati hierarchy is considered. The concept of a generalized solution for such an equation cannot be introduced in the classical theory of generalized functions because the product of generalized functions is not defined. To introduce the concept of a generalized solution, an approach is considered in which the construction of a generalized solution of the second equation of the Riccati hierarchy is carried out using approximation by solutions of the Cauchy problem with complex initial conditions. The total number of generalized solutions depends on the initial conditions of the Cauchy problem, and their form depends on the location of the poles of the approximating solutions.

Détail

Keywords: generalized function, generalized solution of a nonlinear differential equation, Painleve property.

Bibliographie

[1] V.S. Vladimirov, Obobschennye funktsii v matematicheskoi fizike, Nauka, M., 1979, 320 pp.

[2] V.K. Ivanov, “Giperraspredeleniya i umnozhenie raspredelenii Shvartsa”, Doklady Akadademii nauk SSSR, 204:5 (1972), 1045–1048 | Zbl

[3] Yu.V. Egorov, “K teorii obobschennykh funktsii”, Uspekhi matematicheskikh nauk, 45:5 (1990), 3–40

[4] J.F. Colombeau, New generalized functions and multiplication of distributions, North-Holland, Amsterdam, 1984, 374 pp. | MR | Zbl

[5] J.F. Colombeau, “A multiplication of distributions”, Journal of mathematical analysis and applications, 94 (1983), 96–115 | DOI | MR | Zbl

[6] T.G. Shagova, “Ratsionalnye mnemofunktsii na $\mathbb{R}$”, Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika, 2019, no. 2, 6–17

[7] E.L. Ince, Ordinary differential equations, Dover Publications, New York, 1944, 558 pp. | MR | Zbl

[8] V.I. Gromak, N.A. Lukashevich, Analiticheskie svoistva reshenii uravnenii Penleve, Universitetskoe, Minsk, 1990, 157 pp.

[9] V.I. Gromak, “O resheniyakh vtorogo uravneniya Penleve”, Differentsialnye uravneniya, 18:5 (1982), 753–763 | Zbl

[10] E.V. Gritsuk, E.B. Kuzmina, “Issledovanie obobschennoi ierarkhii uravneniya Rikkati na svoistvo Penleve”, Vesnik Brestskaga universiteta. Seryya 4. Fizika. Matematyka, 2017, no. 2, 64–72

[11] S. Albeverio, F. Gestezi, R. Kheeg-Kron, Kh. Kholden, Reshaemye modeli v kvantovoi mekhanike, Mir, M., 1991, 568 pp.

[12] V.G. Danilov, V.P. Maslov, V.M. Shelkovich, “Algebry osobennostei singulyarnykh reshenii kvazilineinykh strogo giperbolicheskikh sistem pervogo poryadka”, TMF, 114:1 (1998), 3–55 | Zbl

[13] A.B. Antonevich, T.G. Shagova, “Obobschennye resheniya odnogo differentsialnogo uravneniya s ratsionalnym koeffitsientom”, Tavricheskii Vestnik Informatiki i Matematiki, 2019, no. 3, 23–36

[14] A.B. Antonevich, E.B. Kuzmina, “Resheniya differentsialnogo uravneniya $u'+\frak s x u=0$ v prostranstve raspredelenii”, Vesnik Grodzenskaga dzyarzhaunaga universiteta imya Ya. Kupaly. Seryya 2. Matematyka. Fizika. infarmatyka, vylichalnaya tekhnika i kiravanne, 10:2 (2020), 56–66

[15] E.B. Kuzmina, “Obobschennye resheniya differentsialnogo uravneniya pervogo poryadka s ratsionalnym koeffitsientom spetsialnogo vida”, Problemy fiziki, matematiki i tekhniki, 2021, no. 1 (46), 54–61