Dynamical canonical systems and their explicit solutions

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March 2017, 37(3): 1679-1689. doi: 10.3934/dcds.2017069

Dynamical canonical systems and their explicit solutions

Alexander Sakhnovich

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

Received April 2016 Revised July 2016 Published December 2016

Fund Project: This research was supported by the Austrian Science Fund (FWF) under Grants No. P24301 and No. P29177.

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.

Keywords: Dynamical canonical system, canonical system, Hamiltonian, Bäcklund-Darboux transformation, explicit solutions, energy, $C_0$-semigroup, well-posedness.

Mathematics Subject Classification: Primary:35B06, 37D99;Secondary:37C80, 74H05.

Citation: Alexander Sakhnovich. Dynamical canonical systems and their explicit solutions. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1679-1689. doi: 10.3934/dcds.2017069

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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