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Renormalizable Expanding Baker Maps: Coexistence of strange attractors
Dynamical canonical systems and their explicit solutions
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria |
Dynamical canonical systems and their connections with the classical (spectral) canonical systems are considered. We construct Bäcklund-Darboux transformation and explicit solutions of the dynamical canonical systems. We study also those properties of the solutions, which are of interest in evolution and control theories.
References:
[1] |
K. R. Acharya, Remling's theorem on canonical systems J. Math. Phys. 57 (2016), 023505, 11 pp.
doi: 10.1063/1.4940048. |
[2] |
M. Belishev and V. Mikhailov, Inverse problem for a one-dimensional dynamical Dirac system (BC-method)
Inverse Problems 30 (2014), 125013, 26pp.
doi: 10.1088/0266-5611/30/12/125013. |
[3] |
L. Bociu, B. Kaltenbacher and P. Radu (eds),
Special volume on nonlinear pdes and control theory with applications, Evol. Equ. Control Theory EECT, 2 (2013), ⅰ-ⅱ.
doi: 10.3934/eect.2013.2.2i. |
[4] |
A. M. Bruckstein and T. Kailath,
Inverse scattering for discrete transmission-line models, SIAM Rev., 29 (1987), 359-389.
doi: 10.1137/1029075. |
[5] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-Time Dynamics Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[6] |
R. M. Colombo,
Well posedness and control in models based on conservation laws, in Nonlinear conservation laws and applications (eds. A. Bressan et al.), IMA Vol. Math. Appl., 153 (2011), 267-277.
doi: 10.1007/978-1-4419-9554-4_13. |
[7] |
L. de Branges,
Hilbert Spaces of Entire Functions Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1968. |
[8] |
P. A. Deift,
Applications of a commutation formula, Duke Math. J., 45 (1978), 267-310.
doi: 10.1215/S0012-7094-78-04516-7. |
[9] |
B. Feng and D. Zhao,
Optimal bilinear control of Gross-Pitaevskii equations with Coulombian potentials, J. Differential Equations, 260 (2016), 2973-2993.
doi: 10.1016/j.jde.2015.10.026. |
[10] |
B. Fritzsche, B. Kirstein, I. Ya. Roitberg and A. L. Sakhnovich, Pseudo-exponential-type solutions of wave equations depending on several variables SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 010, 13 pp.
doi: 10.3842/SIGMA.2015.010. |
[11] |
F. Gesztesy,
A complete spectral characterization of the double commutation method, J. Funct. Anal., 117 (1993), 401-446.
doi: 10.1006/jfan.1993.1132. |
[12] |
F. Gesztesy and G. Teschl,
On the double commutation method, Proc. Amer. Math. Soc., 124 (1996), 1831-1840.
doi: 10.1090/S0002-9939-96-03299-6. |
[13] |
I. Gohberg, M. A. Kaashoek and A. L. Sakhnovich,
Scattering problems for a canonical system with a pseudo-exponential potential, Asymptotic Analysis, 29 (2002), 1-38.
|
[14] |
I. Gohberg and M. G. Krein,
Theory and Applications of Volterra Operators in Hilbert Space Transl. of math. monographs, 24 Amer. Math. Soc. , Providence, RI, 1970. |
[15] |
C. H. Gu, H. Hu and Z. Zhou,
Darboux Transformations in Integrable Systems. Theory and Their Applications to Geometry Mathematical Physics Studies, 26 Springer, Dordrecht, 2005.
doi: 10.1007/1-4020-3088-6. |
[16] |
B. Jacob, K. Morris and H. Zwart,
$C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain, J. Evol. Equ., 15 (2015), 493-502.
doi: 10.1007/s00028-014-0271-1. |
[17] |
A. Kostenko, A. Sakhnovich and G. Teschl,
Commutation methods for Schrödinger operators with strongly singular potentials, Math. Nachr., 285 (2012), 392-410.
doi: 10.1002/mana.201000108. |
[18] |
F. Linares and L. Rosier,
Control and stabilization of the Benjamin-Ono equation on a periodic domain, Trans. Amer. Math. Soc., 367 (2015), 4595-4626.
doi: 10.1090/S0002-9947-2015-06086-3. |
[19] |
V. B. Matveev and M. A. Salle,
Darboux Transformations and Solitons Springer Series in Nonlinear Dynamics, Springer, Berlin, 1991.
doi: 10.1007/978-3-662-00922-2. |
[20] |
R. Mennicken, A. L. Sakhnovich and C. Tretter,
Direct and inverse spectral problem for a system of differential equations depending rationally on the spectral parameter, Duke Math. J., 109 (2001), 413-449.
doi: 10.1215/S0012-7094-01-10931-9. |
[21] |
V. I. Mogilevskii,
Spectral and pseudospectral functions of Hamiltonian systems: Development of the results by Arov-Dym and Sakhnovich, Methods Funct. Anal. Topology, 21 (2015), 370-402.
|
[22] |
L. Molinet and D. Pilod,
Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371.
doi: 10.1016/j.anihpc.2013.12.003. |
[23] |
Rakesh,
A one dimensional inverse problem for a hyperbolic system with complex coefficients, Inverse Problems, 17 (2001), 1401-1417.
doi: 10.1088/0266-5611/17/5/311. |
[24] |
A. L. Sakhnovich,
Dressing procedure for solutions of nonlinear equations and the method of operator identities, Inverse Problems, 10 (1994), 699-710.
doi: 10.1088/0266-5611/10/3/013. |
[25] |
A. L. Sakhnovich,
Iterated Bäcklund-Darboux transform for canonical systems, J. Funct. Anal., 144 (1997), 359-370.
doi: 10.1006/jfan.1996.3003. |
[26] |
A. L. Sakhnovich,
Generalized Bäcklund-Darboux transformation: Spectral properties and nonlinear equations, J. Math. Anal. Appl., 262 (2001), 274-306.
doi: 10.1006/jmaa.2001.7577. |
[27] |
A. L. Sakhnovich,
Dirac type and canonical systems: Spectral and Weyl-Titchmarsh matrix fuctions, direct and inverse problems, Inverse Problems, 18 (2002), 331-348.
doi: 10.1088/0266-5611/18/2/303. |
[28] |
A. L. Sakhnovich,
Dirac type system on the axis: Explicit formulas for matrix potentials with singularities and soliton-positon interactions, Inverse Problems, 19 (2003), 845-854.
doi: 10.1088/0266-5611/19/4/304. |
[29] |
A. L. Sakhnovich,
On the GBDT version of the Bäcklund-Darboux transformation and its applications to linear and nonlinear equations and Weyl theory, Math. Model. Nat. Phenom., 5 (2010), 340-389.
doi: 10.1051/mmnp/20105415. |
[30] |
A. L. Sakhnovich,
Inverse problem for Dirac systems with locally square-summable potentials and rectangular Weyl functions, J. Spectr. Theory, 5 (2015), 547-569.
doi: 10.4171/JST/106. |
[31] |
A. L. Sakhnovich, Dynamical and spectral Dirac systems: Response function and inverse problems J. Math. Phys. 56 (2015), 112702, 13 pp.
doi: 10.1063/1.4936073. |
[32] |
A. L. Sakhnovich, L. A. Sakhnovich and I. Ya. Roitberg,
Inverse Problems and Nonlinear Evolution Equations. Solutions, Darboux Matrices and Weyl-Titchmarsh Functions, De Gruyter Studies in Mathematics,, 47 (2013).
doi: 10.1515/9783110258615. |
[33] |
L. A. Sakhnovich, The factorization of an operator-valued transmission function (Russian),
Dokl. Akad. Nauk SSSR, 226 (1976), 781-784; Translated in: Sov. Math. Dokl. , 17 (1976),
203-207. |
[34] |
L. A. Sakhnovich,
Spectral Theory of Canonical Differential Systems, Method of Operator Identities Operator Theory Adv. Appl. , 107 Birkhäuser, Basel, 1999.
doi: 10.1007/978-3-0348-8713-7. |
[35] |
V. Vasan and B. Deconinck,
Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst., 33 (2013), 3171-3188.
doi: 10.3934/dcds.2013.33.3171. |
[36] |
H. Winkler and H. Woracek,
A growth condition for Hamiltonian systems related with Krein strings, Acta Sci. Math. (Szeged), 80 (2014), 31-94.
doi: 10.14232/actasm-012-028-8. |
[37] |
H. Zwart, Y. Le Gorrec, B. Maschke and J. Villegas,
Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM Control Optim. Calc. Var., 16 (2010), 1077-1093.
doi: 10.1051/cocv/2009036. |
show all references
References:
[1] |
K. R. Acharya, Remling's theorem on canonical systems J. Math. Phys. 57 (2016), 023505, 11 pp.
doi: 10.1063/1.4940048. |
[2] |
M. Belishev and V. Mikhailov, Inverse problem for a one-dimensional dynamical Dirac system (BC-method)
Inverse Problems 30 (2014), 125013, 26pp.
doi: 10.1088/0266-5611/30/12/125013. |
[3] |
L. Bociu, B. Kaltenbacher and P. Radu (eds),
Special volume on nonlinear pdes and control theory with applications, Evol. Equ. Control Theory EECT, 2 (2013), ⅰ-ⅱ.
doi: 10.3934/eect.2013.2.2i. |
[4] |
A. M. Bruckstein and T. Kailath,
Inverse scattering for discrete transmission-line models, SIAM Rev., 29 (1987), 359-389.
doi: 10.1137/1029075. |
[5] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-Time Dynamics Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[6] |
R. M. Colombo,
Well posedness and control in models based on conservation laws, in Nonlinear conservation laws and applications (eds. A. Bressan et al.), IMA Vol. Math. Appl., 153 (2011), 267-277.
doi: 10.1007/978-1-4419-9554-4_13. |
[7] |
L. de Branges,
Hilbert Spaces of Entire Functions Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1968. |
[8] |
P. A. Deift,
Applications of a commutation formula, Duke Math. J., 45 (1978), 267-310.
doi: 10.1215/S0012-7094-78-04516-7. |
[9] |
B. Feng and D. Zhao,
Optimal bilinear control of Gross-Pitaevskii equations with Coulombian potentials, J. Differential Equations, 260 (2016), 2973-2993.
doi: 10.1016/j.jde.2015.10.026. |
[10] |
B. Fritzsche, B. Kirstein, I. Ya. Roitberg and A. L. Sakhnovich, Pseudo-exponential-type solutions of wave equations depending on several variables SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 010, 13 pp.
doi: 10.3842/SIGMA.2015.010. |
[11] |
F. Gesztesy,
A complete spectral characterization of the double commutation method, J. Funct. Anal., 117 (1993), 401-446.
doi: 10.1006/jfan.1993.1132. |
[12] |
F. Gesztesy and G. Teschl,
On the double commutation method, Proc. Amer. Math. Soc., 124 (1996), 1831-1840.
doi: 10.1090/S0002-9939-96-03299-6. |
[13] |
I. Gohberg, M. A. Kaashoek and A. L. Sakhnovich,
Scattering problems for a canonical system with a pseudo-exponential potential, Asymptotic Analysis, 29 (2002), 1-38.
|
[14] |
I. Gohberg and M. G. Krein,
Theory and Applications of Volterra Operators in Hilbert Space Transl. of math. monographs, 24 Amer. Math. Soc. , Providence, RI, 1970. |
[15] |
C. H. Gu, H. Hu and Z. Zhou,
Darboux Transformations in Integrable Systems. Theory and Their Applications to Geometry Mathematical Physics Studies, 26 Springer, Dordrecht, 2005.
doi: 10.1007/1-4020-3088-6. |
[16] |
B. Jacob, K. Morris and H. Zwart,
$C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain, J. Evol. Equ., 15 (2015), 493-502.
doi: 10.1007/s00028-014-0271-1. |
[17] |
A. Kostenko, A. Sakhnovich and G. Teschl,
Commutation methods for Schrödinger operators with strongly singular potentials, Math. Nachr., 285 (2012), 392-410.
doi: 10.1002/mana.201000108. |
[18] |
F. Linares and L. Rosier,
Control and stabilization of the Benjamin-Ono equation on a periodic domain, Trans. Amer. Math. Soc., 367 (2015), 4595-4626.
doi: 10.1090/S0002-9947-2015-06086-3. |
[19] |
V. B. Matveev and M. A. Salle,
Darboux Transformations and Solitons Springer Series in Nonlinear Dynamics, Springer, Berlin, 1991.
doi: 10.1007/978-3-662-00922-2. |
[20] |
R. Mennicken, A. L. Sakhnovich and C. Tretter,
Direct and inverse spectral problem for a system of differential equations depending rationally on the spectral parameter, Duke Math. J., 109 (2001), 413-449.
doi: 10.1215/S0012-7094-01-10931-9. |
[21] |
V. I. Mogilevskii,
Spectral and pseudospectral functions of Hamiltonian systems: Development of the results by Arov-Dym and Sakhnovich, Methods Funct. Anal. Topology, 21 (2015), 370-402.
|
[22] |
L. Molinet and D. Pilod,
Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371.
doi: 10.1016/j.anihpc.2013.12.003. |
[23] |
Rakesh,
A one dimensional inverse problem for a hyperbolic system with complex coefficients, Inverse Problems, 17 (2001), 1401-1417.
doi: 10.1088/0266-5611/17/5/311. |
[24] |
A. L. Sakhnovich,
Dressing procedure for solutions of nonlinear equations and the method of operator identities, Inverse Problems, 10 (1994), 699-710.
doi: 10.1088/0266-5611/10/3/013. |
[25] |
A. L. Sakhnovich,
Iterated Bäcklund-Darboux transform for canonical systems, J. Funct. Anal., 144 (1997), 359-370.
doi: 10.1006/jfan.1996.3003. |
[26] |
A. L. Sakhnovich,
Generalized Bäcklund-Darboux transformation: Spectral properties and nonlinear equations, J. Math. Anal. Appl., 262 (2001), 274-306.
doi: 10.1006/jmaa.2001.7577. |
[27] |
A. L. Sakhnovich,
Dirac type and canonical systems: Spectral and Weyl-Titchmarsh matrix fuctions, direct and inverse problems, Inverse Problems, 18 (2002), 331-348.
doi: 10.1088/0266-5611/18/2/303. |
[28] |
A. L. Sakhnovich,
Dirac type system on the axis: Explicit formulas for matrix potentials with singularities and soliton-positon interactions, Inverse Problems, 19 (2003), 845-854.
doi: 10.1088/0266-5611/19/4/304. |
[29] |
A. L. Sakhnovich,
On the GBDT version of the Bäcklund-Darboux transformation and its applications to linear and nonlinear equations and Weyl theory, Math. Model. Nat. Phenom., 5 (2010), 340-389.
doi: 10.1051/mmnp/20105415. |
[30] |
A. L. Sakhnovich,
Inverse problem for Dirac systems with locally square-summable potentials and rectangular Weyl functions, J. Spectr. Theory, 5 (2015), 547-569.
doi: 10.4171/JST/106. |
[31] |
A. L. Sakhnovich, Dynamical and spectral Dirac systems: Response function and inverse problems J. Math. Phys. 56 (2015), 112702, 13 pp.
doi: 10.1063/1.4936073. |
[32] |
A. L. Sakhnovich, L. A. Sakhnovich and I. Ya. Roitberg,
Inverse Problems and Nonlinear Evolution Equations. Solutions, Darboux Matrices and Weyl-Titchmarsh Functions, De Gruyter Studies in Mathematics,, 47 (2013).
doi: 10.1515/9783110258615. |
[33] |
L. A. Sakhnovich, The factorization of an operator-valued transmission function (Russian),
Dokl. Akad. Nauk SSSR, 226 (1976), 781-784; Translated in: Sov. Math. Dokl. , 17 (1976),
203-207. |
[34] |
L. A. Sakhnovich,
Spectral Theory of Canonical Differential Systems, Method of Operator Identities Operator Theory Adv. Appl. , 107 Birkhäuser, Basel, 1999.
doi: 10.1007/978-3-0348-8713-7. |
[35] |
V. Vasan and B. Deconinck,
Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst., 33 (2013), 3171-3188.
doi: 10.3934/dcds.2013.33.3171. |
[36] |
H. Winkler and H. Woracek,
A growth condition for Hamiltonian systems related with Krein strings, Acta Sci. Math. (Szeged), 80 (2014), 31-94.
doi: 10.14232/actasm-012-028-8. |
[37] |
H. Zwart, Y. Le Gorrec, B. Maschke and J. Villegas,
Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM Control Optim. Calc. Var., 16 (2010), 1077-1093.
doi: 10.1051/cocv/2009036. |
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