Geodesic
Evolution of Weyl Functions and Initial-Boundary Value Problems
A. L. Sakhnovich1
1 Vienna University of Technology, Austria
Mathematical modelling of natural phenomena, Tome 11 (2016) no. 2, pp. 111-132

Voir la notice de l'article provenant de la source EDP Sciences

This review is dedicated to some recent results on Weyl theory, inverse problems, evolution of the Weyl functions and applications to integrable wave equations in a semistrip and quarter-plane. For overdetermined initial-boundary value problems, we consider some approaches, which help to reduce the number of the initial-boundary conditions. The interconnections between dynamical and spectral Dirac systems, between response and Weyl functions are studied as well.
DOI : 10.1051/mmnp/201611209

A. L. Sakhnovich 1

1 Vienna University of Technology, Austria 
@article{10_1051_mmnp_201611209,
     author = {A. L. Sakhnovich},
     title = {Evolution of {Weyl} {Functions} and {Initial-Boundary} {Value} {Problems}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {111--132},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
     year = {2016},
     doi = {10.1051/mmnp/201611209},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/201611209/}
}
TY  - JOUR
AU  - A. L. Sakhnovich
TI  - Evolution of Weyl Functions and Initial-Boundary Value Problems
JO  - Mathematical modelling of natural phenomena
PY  - 2016
SP  - 111
EP  - 132
VL  - 11
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/201611209/
DO  - 10.1051/mmnp/201611209
LA  - en
ID  - 10_1051_mmnp_201611209
ER  - 
%0 Journal Article
%A A. L. Sakhnovich
%T Evolution of Weyl Functions and Initial-Boundary Value Problems
%J Mathematical modelling of natural phenomena
%D 2016
%P 111-132
%V 11
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/201611209/
%R 10.1051/mmnp/201611209
%G en
%F 10_1051_mmnp_201611209
A. L. Sakhnovich. Evolution of Weyl Functions and Initial-Boundary Value Problems. Mathematical modelling of natural phenomena, Tome 11 (2016) no. 2, pp. 111-132. doi: 10.1051/mmnp/201611209

[1] M.J. Ablowitz, R. Haberman J. Math. Phys. 1975 2301 2305

[2] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur Phys. Rev. Lett. 1973 1262 1264

[3] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur Stud. Appl. Math. 1974 249 315

[4] S. Albeverio, R. Hryniv, Ya. Mykytyuk J. Math. Anal. Appl. 2008 45 57

[5] A.C. Ashton Proc. R. Soc. A 2012 1325 1331

[6] S. Avdonin, M. Belishev, S. Ivanov Boundary control and an inverse matrix problem for the equation utt − uxx + V(x)u Math. USSR-Sb. 1992 287 310

[7] S. Avdonin, V. Mikhaylov, K. Ramdani IMA J. Math. Control Inform. 2014 137 150

[8] S. Avdonin, V. Mikhaylov, A. Rybkin Comm. Math. Phys. 2007 791 803

[9] T. Bang. On quasi-analytic functions. In: C. R. Dixieme Congres Math. Scandinaves 1946. Jul. Gjellerups Forlag, Copenhagen, 1947, pp. 249–254.

[10] I.V. Barashenkov, D.E. Pelinovsky Phys. Lett. B 1998 117 124

[11] R. Beals, R.R. Coifman Comm. Pure Appl. Math. 1984 39 90

[12] M. Belishev Math. USSR-Sb. 1990 23 42

[13] M. Belishev. Boundary control method in dynamical inverse problems–an introductory course. In: Dynamical Inverse Problems: Theory and Application. CISM Courses and Lectures, Vol. 529, Springer, Vienna, 2011, pp. 85–150.

[14] M. Belishev, V. Mikhailov Inverse Problems 2014 125013

[15] Yu.M. Berezanskii Dokl. Akad. Nauk SSSR 1985 16 19

[16] A. Beurling. Mittag–Leffler lectures on complex analysis. In: L. Carleson, P. Malliavin, J. Neuberger, J. Wermer (eds). The Collected Works of Arne Beurling. Vol. 1: Complex Analysis. Contemporary Mathematicians, Birkhäuser, Boston, MA, 1989, pp. 361–444.

[17] A.S. Blagoveščenskii The local method of solution of the nonstationary inverse problem for an inhomogeneous string (Russian) Trudy Mat. Inst. Steklov. 1971 28 38

[18] D. Bollé, F. Gesztesy, H. Grosse, W. Schweiger, B. Simon J. Math. Phys. 1987 1512 1525

[19] J.L. Bona, A.S. Fokas Nonlinearity 2008 T195 T203

[20] P. Bowcock, G. Tzamtzis J. High Energy Phys. 2007 22

[21] R. Brunnhuber, J. Eckhardt, A. Kostenko, G. Teschl Monatsh. Math. 2014 515 547

[22] S. Clark, F. Gesztesy Contemp. Math. 2006 103 140

[23] S. Clark, F. Gesztesy, W. Renger J. Differential Equations 2005 144 182

[24] J. Eckhardt, F. Gesztesy, R. Nichols, G. Teschl J. Spectr. Theory 2014 715 768

[25] J. Eckhardt, F. Gesztesy, R. Nichols, A. Sakhnovich, G. Teschl Differential Integral Equations 2015 505 522

[26] L.D. Faddeev, L.A. Takhtajan. Hamiltonian Methods in the Theory of Solitons. Springer, Berlin, 1987.

[27] A.S. Fokas Comm. Math. Phys. 2002 1 39

[28] A.S. Fokas. A Unified Approach to Boundary Value Problems. CBMS-NSF Regional Conference Ser. in Appl. Math. vol. 78. SIAM, Philadelphia, 2008.

[29] B. Fritzsche, B. Kirstein, I.Ya. Roitberg, A.L. Sakhnovich Integral Equations Operator Theory 2012 163 187

[30] B. Fritzsche, B. Kirstein, I.Ya. Roitberg, A.L. Sakhnovich Oper. Matrices 2013 183 196

[31] F. Gesztesy, J.A. Goldstein, H. Holden, G. Teschl Ann. Mat. Pura Appl. 2012 631 676

[32] F. Gesztesy, W. Schweiger, B. Simon Trans. Amer. Math. Soc. 1991 465 525

[33] F. Gesztesy, B. Simon Ann. of Math. (2) 2000 593 643

[34] G.M.L. Gladwell, A. Morassi (eds). Dynamical Inverse Problems: Theory and Application. CISM Courses and Lectures, vol. 529. Springer, Vienna, 2011.

[35] R.O. Hryniv J. Math. Phys. 2011 063513

[36] M. Kac, P. Van Moerbeke Proc. Natl. Acad. Sci. USA 1975 2879 2880

[37] S. Kamvissis, D. Shepelsky, L. Zielinski J. Nonlinear Math. Phys. 2015 448 473

[38] V.G. Khryptun Ukrainian Math. J. 1989 569 584

[39] M.G. Krein Continuous analogues of propositions on polynomials orthogonal on the unit circle (Russian) Dokl. Akad. Nauk SSSR 1955 637 640

[40] I.M. Krichever Dokl. Akad. Nauk. SSSR 1980 288 292

[41] B.M. Levitan, I.S. Sargsjan. Introduction to the Spectral Theory. Selfadjoint Differential Operators. Transl. Math. Monographs, vol. 34. AMS, Providence, RI, 1975.

[42] A.N. Leznov. M.V. Saveliev. Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems. Progress in Physics, vol 15. Birkhäuser, Basel, 1992.

[43] F. Lund, T. Regge Phys. Rev. D (3) 1976 1524 1535

[44] A. Mercado, A. Osses, L. Rosier Inverse Problems 2008 015017

[45] Ya.V. Mykytyuk, D.V. Puyda J. Math. Anal. Appl. 2012 177 194

[46] S.P. Novikov Funct. Anal. Appl. 1974 236 246

[47] K. Pohlmeyer Commun. Math. Phys. 1976 207 221

[48] A.L. Sakhnovich Ukr. Math. J. 1990 316 323

[49] A.L. Sakhnovich Russ. Math. Surveys 1991 198 200

[50] A.L. Sakhnovich Russ. Math. Iz. VUZ 1992 42 52

[51] A.L. Sakhnovich Inverse Problems 2002 331 348

[52] A.L. Sakhnovich Inverse Problems 2005 703 716

[53] A.L. Sakhnovich Inverse Problems 2008 23

[54] A.L. Sakhnovich Math. Model. Nat. Phenom. 2012 131 145

[55] A.L. Sakhnovich J. Differential Equations 2012 3658 3667

[56] A.L. Sakhnovich J. Spectr. Theory 2015 547 569

[57] A.L. Sakhnovich SIGMA Symmetry Integrability Geom. Methods Appl. 2016 17

[58] A.L. Sakhnovich J. Math. Anal. Appl. 2015 746 757

[59] A.L. Sakhnovich J. Math. Phys. 2015

[60] A.L. Sakhnovich, L.A. Sakhnovich, I.Ya. Roitberg. Inverse Problems and Nonlinear Evolution Equations. Solutions, Darboux Matrices and Weyl–Titchmarsh Functions. De Gruyter, Berlin, 2013.

[61] L.A. Sakhnovich. The Non-Linear Equations and the Inverse Problems on the Half-Axis. Preprint 30. Inst. Matem. AN Ukr.SSR, Kiev, 1987.

[62] L.A. Sakhnovich St. Petersburg Math. J. 1994 1 69

[63] L.A. Sakhnovich. Spectral Theory of Canonical Differential Systems. Method of Operator Identities. Operator Theory Adv. Appl., vol. 107. Birkhäuser, Basel, 1999.

[64] V.E. Zakharov, S.V. Manakov Soviet Physics JETP 1975 1654 1673

[65] V.E. Zakharov, A.V. Mikhailov Soviet Physics JETP 1978 1017 1027

[66] V.E. Zakharov, A.B. Shabat Soviet Physics JETP 1972 62 69

[67] V.E. Zakharov, A.B. Shabat II. Funct. Anal. Appl. 1979 166 174

Cité par Sources :