Geodesic
Singularities of quasi-linear differential equations
Dalʹnevostočnyj matematičeskij žurnal, Tome 23 (2023) no. 1, pp. 85-105.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study solutions of quasi-linear ordinary differential equations of the second order at their singular points, where the coefficient of the second-order derivative vanishes. Either solutions entering a singular point with definite tangential direction (proper solutions) or those without definite tangential direction (oscillating solutions) are considered. It is shown that oscillating solutions generically do not exist, and proper solutions enter a singular point in strictly definite tangential directions. A local representation for proper solutions in a form similar to Newton–Puiseux series is obtained. 
@article{DVMG_2023_23_1_a7,
     author = {A. O. Remizov},
     title = {Singularities of quasi-linear differential equations},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {85--105},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2023_23_1_a7/}
}
TY  - JOUR
AU  - A. O. Remizov
TI  - Singularities of quasi-linear differential equations
JO  - Dalʹnevostočnyj matematičeskij žurnal
PY  - 2023
SP  - 85
EP  - 105
VL  - 23
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DVMG_2023_23_1_a7/
LA  - ru
ID  - DVMG_2023_23_1_a7
ER  - 
%0 Journal Article
%A A. O. Remizov
%T Singularities of quasi-linear differential equations
%J Dalʹnevostočnyj matematičeskij žurnal
%D 2023
%P 85-105
%V 23
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DVMG_2023_23_1_a7/
%G ru
%F DVMG_2023_23_1_a7
A. O. Remizov. Singularities of quasi-linear differential equations. Dalʹnevostočnyj matematičeskij žurnal, Tome 23 (2023) no. 1, pp. 85-105. http://geodesic.mathdoc.fr/item/DVMG_2023_23_1_a7/

[1] A. F. Filippov, “Edinstvennost resheniya sistemy differentsialnykh uravnenii, ne razreshennykh otnositelno proizvodnykh”, Differents. uravn., 41:1 (2005), 87–92 | MR | Zbl

[2] R. Lamour, R. Marz, C. Tischendorf, Differential-algebraic equations. A projector based analysis, Springer, Berlin, 2013 | MR | Zbl

[3] J. Sotomayor, M. Zhitomirskii, “Impasse singularities of differential systems of the form $A(x)x' = F(x)$”, J. Differ. Equations, 169:2 (2001), 567–587 | DOI | MR | Zbl

[4] R. Ignat, L. Nguyen, V. Slastikov, A. Zarnescu, “Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals”, SIAM J. Math. Anal., 46 (2014), 3390–3425 | DOI | MR | Zbl

[5] L. V. Lokutsievskiy, M. I. Zelikin, “The analytical solution of Newton's aerodynamic problem in the class of bodies with vertical plane of symmetry and developable side boundary”, ESAIM, Control Optim. Calc. Var., 26 (2020), 15, 36 pp. | DOI | MR | Zbl

[6] A. V. Aminova, N. A.-M. Aminov, “Proektivnaya geometriya sistem differentsialnykh uravnenii vtorogo poryadka”, Matem. sb., 197:7 (2006), 3–28 | DOI | MR | Zbl

[7] V. A. Yumaguzhin, “Differential invariants of second order ODEs. I”, Acta Appl. Math., 109:1 (2010), 283–313 | DOI | MR | Zbl

[8] A. O. Remizov, “Geodezicheskie na dvumernykh poverkhnostyakh s psevdorimanovoi metrikoi: osobennosti smeny signatury”, Matem. sb., 200:3 (2009), 75–94 | DOI | MR | Zbl

[9] A. O. Remizov, “O geodezicheskikh v metrikakh s osobennostyami tipa Kleina”, UMN, 65:1 (2010), 187–188 | DOI | MR | Zbl

[10] A. O. Remizov, “On the local and global properties of geodesics in pseudo-Riemannian metrics”, Differ. Geom. Appl., 39 (2015), 36–58 | DOI | MR | Zbl

[11] A. O. Remizov, F. Tari, “Singularities of the geodesic flow on surfaces with pseudo-Riemannian metrics”, Geom. Dedicata, 185:1 (2016), 131–153 | DOI | MR | Zbl

[12] N. G. Pavlova, A. O. Remizov, “A brief survey on singularities of geodesic flows in smooth signature changing metrics on 2-surfaces”, Singularities and foliations. Geometry, topology and applications, Springer Proc. Math. Stat., 222, Springer, Cham, 2018, 135–155 | MR | Zbl

[13] N. G. Pavlova, A. O. Remizov, “Zavershenie klassifikatsii tipichnykh osobennostei geodezicheskikh potokov v metrikakh dvukh klassov”, Izv. RAN. Ser. matem., 83:1 (2019), 119–139 | DOI | MR | Zbl

[14] A. Honda, K. Saji, K. Teramoto, “Mixed type surfaces with bounded Gaussian curvature in three-dimensional Lorentzian manifolds”, Adv. Math., 365 (2020), 107036, 46 pp. | DOI | MR | Zbl

[15] I. A. Bogaevskii, D. V. Tunitskii, “Osobennosti mnogoznachnykh reshenii kvazilineinykh giperbolicheskikh sistem”, Trudy MIAN, 308 (2020), 76–87 | DOI | Zbl

[16] A. D. Bryuno, Lokalnyi metod nelineinogo analiza differentsialnykh uravnenii, Nauka, M., 1979 | MR

[17] A. D. Bryuno, “Asimptotiki i razlozheniya reshenii obyknovennogo differentsialnogo uravneniya”, UMN, 59:3 (2004), 31–80 | DOI | MR | Zbl

[18] A. A. Davydov, “Normalnaya forma differentsialnogo uravneniya, ne razreshennogo otnositelno proizvodnoi, v okrestnosti ego osoboi tochki”, Funkts. analiz i ego pril., 19:2 (1985), 1–10 | Zbl

[19] A. A. Davydov, G. Ishikawa, S. Izumiya, W.-Z. Sun, “Generic singularities of implicit systems of first order differential equations on the plane”, Japanese J. Math. 3rd Ser., 3:1 (2008), 93–119 | DOI | MR | Zbl

[20] S. Izumiya, W.-Z. Sun, “Singularities of solution surfaces for quasilinear first-order partial differential equations”, Geom. Dedicata, 64:3 (1997), 331–341 | DOI | MR | Zbl

[21] S. Izumiya, F. Tari, “Self-adjoint operators on surfaces with singular metrics”, JDCS, 16:3 (2010), 329–353 | MR | Zbl

[22] M. Lange-Hegermann, D. Robertz, W. M. Seiler, M. Seiss, “Singularities of algebraic differential equations”, Adv. Appl. Math., 131 (2021), 102266, 56 pp. | DOI | MR | Zbl

[23] J. Liang, “A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation”, J. Differ. Equations, 246:2 (2009), 819–844 | DOI | MR | Zbl

[24] L. Ortiz-Bobadilla, E. Rosales-González, S. M. Voronin, “Analytic classification of foliations induced by germs of holomorphic vector fields in $(\mathbb{C}^n,0)$ with non-isolated singularities”, JDCS, 25:3 (2019), 491–516 | MR | Zbl

[25] A. O. Remizov, “Multidimensional Poincaré construction and singularities of lifted fields for implicit differential equations”, J. Math. Sci., 151:6 (2008), 3561–3602 | DOI | MR | Zbl

[26] A. O. Remizov, “Geodesics in generalized Finsler spaces: singularities in dimension two”, J. Singul., 14 (2016), 172–193 | MR | Zbl

[27] W. M. Seiler, M. Seiss, “Singular initial value problems for scalar quasi-linear ordinary differential equations”, J. Differ. Equations, 281 (2021), 258–288 | DOI | MR | Zbl

[28] R. Ghezzi, A. O. Remizov, “On a class of vector fields with discontinuities of divide-by-zero type and its applications to geodesics in singular metrics”, JDCS, 18:1 (2012), 135–158 | MR | Zbl

[29] V. I. Arnold, Yu. S. Ilyashenko, “Obyknovennye differentsialnye uravneniya.”, Itogi nauki i tekhniki. Sovrem. probl. mat. Fundam. napravl., 1, VINITI, M., 1985, 7–140

[30] M. W. Hirsch, C. C. Pugh, M. Shub, Invariant manifolds, Lect. Notes Math., 583, Springer-Verlag, Berlin, 1977 | DOI | MR | Zbl

[31] V. S. Samovol, “Ekvivalentnost sistem differentsialnykh uravnenii v okrestnosti osoboi tochki”, Tr. MMO, 44 (1982), 213–234 | MR | Zbl

[32] V. S. Samovol, “Kriterii $C^1$-gladkoi linearizatsii avtonomnoi sistemy v okrestnosti nevyrozhdennoi osoboi tochki”, Matem. zametki, 49:3 (1991), 91–96 | MR

[33] R. Roussarie, “Modèles locaux de champs et de formes”, Asterisque, 30 (1975), 1–181 | MR

[34] N. G. Pavlova, A. O. Remizov, “Giperbolicheskie polya Russari s vyrozhdennoi kvadratichnoi chastyu”, UMN, 76:2 (2021), 183–184 | DOI | MR | Zbl

[35] N. G. Pavlova, A. O. Remizov, “Smooth local normal forms of hyperbolic Roussarie vector fields”, Moscow Math. J., 21:2 (2021), 413–426 | DOI | MR | Zbl

[36] Yu. S. Ilyashenko, S. Yu. Yakovenko, “Konechno-gladkie normalnye formy lokalnykh semeistv diffeomorfizmov i vektornykh polei”, UMN, 46:1 (1991), 3–39 | MR