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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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A. M. Bruckstein and T. Kailath, Inverse scattering for discrete transmission-line models, SIAM Rev., 29 (1987), 359-389.  doi: 10.1137/1029075.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

L. de Branges, Hilbert Spaces of Entire Functions Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1968.  Google Scholar

[8]

P. A. Deift, Applications of a commutation formula, Duke Math. J., 45 (1978), 267-310.  doi: 10.1215/S0012-7094-78-04516-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

A. L. Sakhnovich, Iterated Bäcklund-Darboux transform for canonical systems, J. Funct. Anal., 144 (1997), 359-370.  doi: 10.1006/jfan.1996.3003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. M. Bruckstein and T. Kailath, Inverse scattering for discrete transmission-line models, SIAM Rev., 29 (1987), 359-389.  doi: 10.1137/1029075.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

L. de Branges, Hilbert Spaces of Entire Functions Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1968.  Google Scholar

[8]

P. A. Deift, Applications of a commutation formula, Duke Math. J., 45 (1978), 267-310.  doi: 10.1215/S0012-7094-78-04516-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

A. L. Sakhnovich, Iterated Bäcklund-Darboux transform for canonical systems, J. Funct. Anal., 144 (1997), 359-370.  doi: 10.1006/jfan.1996.3003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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