On Structural Stability of Characteristic Nets and the Cauchy Problem for a Tricomi--Cibrario Type Equation

A. A. Davydov ; Yu. A. Kasten

Informatics and Automation, Optimal control and differential equations, Tome 304 (2019), pp. 159-166.

Résumé

For a generic second-order linear partial differential equation on the plane, the problem of nonlocal normal forms and invariants of a family of its characteristics, as well as the current state of the corresponding theory, is discussed. Potential applications of this theory are demonstrated by the example of solving a special Cauchy problem for a mixed-type equation.

Bibliographie

[1] A. A. Andronov and L. S. Pontryagin, “Structurally stable systems”, Dokl. Akad. Nauk SSSR, 14:5 (1937), 247–250 | MR

[2] V. I. Arnold, Additional Chapters of the Theory of Ordinary Differential Equations, Nauka, Moscow, 1978 | MR | Zbl

[3] Geometrical Methods in the Theory of Ordinary Differential Equations, Grundl. Math. Wiss., 250, Springer, New York, 1983 | MR | Zbl

[4] V. I. Arnold and Yu. S. Ilyashenko, “Ordinary differential equations”, Dynamical Systems–1, Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl., 1, VINITI, Moscow, 1985, 7–140

[5] Dynamical Systems I, Encycl. Math. Sci., 1, Springer, Berlin, 1988, 1–148

[6] V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps: Classification of Critical Points, Caustics and Wave Fronts, Nauka, Moscow, 1982 | MR | Zbl

[7] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps, v. 1, Monogr. Math., 82, Birkhäuser, Boston, 1985 | MR | Zbl

[8] N. I. Bakievich, “Dirichlet problem in a ring for a mixed-type equation”, Izv. Vyssh. Uchebn. Zaved., Mat., 1968, no. 7, 3–9 | MR | Zbl

[9] I. A. Bogaevsky, “Implicit ordinary differential equations: Bifurcations and sharpening of equivalence”, Izv. Math., 78:6 (2014), 1063–1078 | DOI | DOI | MR | Zbl

[10] Bogaevsky I., “Difference between smooth and formal classifications of Morse transitions of implicit ODE”, Real and complex dynamical systems, Int. Conf. on the occasion of Prof. Yu. Ilyashenko's 75th birthday (Moscow, Nov. 26–30, 2018), Moscow, 2018, 43–44

[11] S. A. Chaplygin, On Gas Jets, Gostekhizdat, Moscow, 1949 (in Russian) | MR

[12] Cibrario M., “Sulla riduzione a forma canonica delle equazioni lineari alle derivate parziali di secondo ordine di tipo misto”, Ist. Lombardo. Rend. II. Ser., 65 (1932), 889–906 | MR

[13] A. A. Davydov, “Normal form of a differential equation, not solvable for the derivative, in a neighborhood of a singular point”, Funct. Anal. Appl., 19:2 (1985), 81–89 | DOI | MR | Zbl

[14] A. A. Davydov, “Structural stability of control systems on orientable surfaces”, Math. USSR, Sb., 72:1 (1992), 1–28 | DOI | MR | Zbl | Zbl

[15] Davydov A., “Normal forms of linear second order partial differential equations on the plane”, Sci. China. Math., 61:11 (2018), 1947–1962 | DOI | MR | Zbl

[16] A. A. Davydov and E. Rosales-Gonsales, “The complete classification of typical second-order linear partial-differential equations on the plane”, Dokl. Math., 54:2 (1996), 669–672 | MR | Zbl

[17] Davydov A.A., Rosales-González E., “Smooth normal forms of folded resonance saddles and nodes and complete classification of generic linear second order PDE's on the plane”, EQUADIFF 95: Int. Conf. on differential equations (Lisboa, 1995), eds. by L. Magalhães, C. Rocha, L. Sanchez, Singapore, World Scientific, 1998, 59–68 | MR

[18] A. A. Davydov and L. Trinh Thi Diep, “Normal forms for families of linear equations of mixed type near non-resonant folded singular points”, Russ. Math. Surv., 65:5 (2010), 984–986 | DOI | DOI | MR | Zbl

[19] Davydov A., Trinh Thi Diep L., “Reduction theorem and normal forms of linear second order mixed type PDE families in the plane”, TWMS J. Pure Appl. Math., 2:1 (2011), 44–53 | MR | Zbl

[20] DeBaggis H.F., “Dynamical systems with stable structures”, Contributions to the theory of nonlinear oscillations, v. 2, Ann. Math. Stud., 29, Princeton Univ. Press, Princeton, 1952, 37–59 | MR

[21] Garcia R., Gutierrez C., Sotomayor J., “Structural stability of asymptotic lines on surfaces immersed in $\mathbb R^3$”, Bull. sci. math., 123:8 (1999), 599–622 | DOI | MR

[22] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer, New York, 1973 | MR | MR | Zbl

[23] Yu. A. Grishina and A. A. Davydov, “Structural stability of simplest dynamical inequalities”, Proc. Steklov Inst. Math., 256 (2007), 89–101 | Zbl

[24] I. Ivanova-Karatopraklieva and I. Kh. Sabitov, “Bending of surfaces: Part II”, Geometry–1, Itogi Nauki Tekh., Ser.: Sovrem. Mat. Prilozh., Temat. Obzory, 8, VINITI, Moscow, 1995), 108–167 | DOI | MR | Zbl

[25] J. Math. Sci., 74:3 (1995), 997–1043 | DOI | MR | Zbl

[26] Kasten J.A., “Solvability of the boundary value problem for a Tricomi type equation in the exterior of a disk”, J. Math. Sci., 188:3 (2013), 268–272 | DOI | MR | Zbl

[27] J. Palis, Jr., and W. de Melo, Geometric Theory of Dynamical Systems: An Introduction, Springer, New York, 1982 | MR | Zbl

[28] Peixoto M.M., “Structural stability on two-dimensional manifolds”, Topology, 1:2 (1962), 101–120 | DOI | MR | Zbl

[29] Peixoto M.M., “Structural stability on two-dimensional manifolds: A further remark”, Topology, 2:1-2 (1963), 179–180 | DOI | MR | Zbl

[30] A. D. Piliya and V. I. Fedorov, “Singularities of the field of an electromagnetic wave in a cold anisotropic plasma with two-dimensional inhomogeneity”, Sov. Phys. JETP, 33:1 (1971), 210–215

[31] S. L. Sobolev, Equations of Mathematical Physics, 4th ed., Nauka, Moscow, 1966 | MR | MR | Zbl

[32] Partial Differential Equations of Mathematical Physics, Pergamon, Oxford, 1964 | MR | MR | Zbl

[33] Tricomi F., “Sulle equazioni lineari alle derivate parziali di $2^\circ $ ordine, di tipo misto”, R. Accad. Lincei. Mem. Ser. 5, 14 (1923), 133–247 | Zbl