Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-17T19:37:35.505Z Has data issue: false hasContentIssue false
  • English
  • Français

On the GBDT Version of the Bäcklund-Darboux Transformation andits Applications to Linear and Nonlinear Equations and Weyl Theory

Published online by Cambridge University Press:  12 May 2010

A. Sakhnovich
A. Sakhnovich*
Affiliation:
Department of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria
*
*E-mail: al_sakhnov@yahoo.com
Get access

Abstract

A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systemsdepending rationally on the spectral parameter is treated and its applications tononlinear equations are given. Explicit solutions of direct and inverse problems forDirac-type systems, including systems with singularities, and for the system auxiliary tothe N-wave equation are reviewed. New results on explicit construction ofthe wave functions for radial Dirac equation are obtained.

Keywords

Bäcklund-Darboux transformationWeyl functionreflection coefficientdirect probleminverse problemDirac-type systemradial Dirac equationintegrable equation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 340 - 389
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

To the memory of M.Sh. Birman, with deep respect

References

Ablowitz, M.J., Chakravarty, S., Trubatch, A.D., Villarroel, J.. A novel class of solutions of the non-stationary Schrödinger and the Kadomtsev-Petviashvili I equations . Phys. Lett. A, n267 (2000), No. 2-3, 132–146.CrossRefGoogle Scholar
Ablowitz, M.J., Haberman, R.. Resonantly coupled nonlinear evolution equations . J. Math. Phys., 16 (1975), 2301–2305.CrossRefGoogle Scholar
Adler, M., van Moerbeke, P.. Birkhoff strata, Bäcklund transformations, and regularization of isospectral operators . Adv. Math., 108 (1994), No. 1, 140–204.CrossRefGoogle Scholar
S. Albeverio, R. Hryniv, Ya. Mykytyuk. Reconstruction of radial Dirac operators.J. Math. Phys. 48 (2007), No. 4, 043501, 14 pp.
Alpay, D., Gohberg, I.. Inverse spectral problem for differential operators with rational scattering matrix functions. J. Diff. Eqs., 118 (1995), 1–19. CrossRefGoogle Scholar
Alpay, D., Gohberg, I., Kaashoek, M.A., Sakhnovich, A.L.. Direct and inverse scattering problem for canonical systems with a strictly pseudo-exponential potential . Math. Nachr., 215 (2000), 5–31.3.0.CO;2-M>CrossRefGoogle Scholar
Bäcklund, A.V.. Zur Theorie der partiellen Differential gleichungen erster Ordnung . Math. Ann., 17 (1880), 285–328.CrossRefGoogle Scholar
H. Bart, I. Gohberg, M.A. Kaashoek.Minimal factorization of matrix and operator functions. Operator Theory: Adv. Appl., 1, Birkhäuser Verlag, Basel, 1979.
Beals, R., Coifman, R.R.. Scattering and inverse scattering for first-order systems: II . Inverse Probl., 3 (1987), 577–593.CrossRefGoogle Scholar
Borisov, A.B., Kiseliev, V.V.. Inverse problem for an elliptic sine-Gordon equation with an asymptotic behaviour of the cnoidal-wave type . Inverse Probl., 5 (1989), 959–982.CrossRefGoogle Scholar
Boutet de Monvel, A., Marchenko, V.. Generalization of the Darboux transform . Matematicheskaya fizika, analiz, geometriya, 1 (1994), 479–504. Google Scholar
Carl, B., Schiebold, C.. Nonlinear equations in soliton physics and operator ideals . Nonlinearity, 12 (1999), 333–364.CrossRefGoogle Scholar
Cascaval, R.C., Gesztesy, F., Holden, H., Latushkin, Yu.. Spectral analysis of Darboux transformations for the focusing NLS hierarchy . J. Anal. Math., 93 (2004), 139–197.CrossRefGoogle Scholar
Chudnovsky, D.V., Chudnovsky, G.V.. Bäcklund transformation as a method of decomposition and reproduction of two-dimensional nonlinear systems . Phys. Lett. A, 87 (1982), No. 7, 325–329.CrossRefGoogle Scholar
Cieslinski, J.. An effective method to compute N -fold Darboux matrix and N -soliton surfaces . J. Math. Phys., 32 (1991), 2395–2399.CrossRefGoogle Scholar
Clark, S., Gesztesy, F.. On self-adjoint and J -self-adjoint Dirac-type operators: a case study . Contemporary Mathematics, 412 (2006), 103–140.CrossRefGoogle Scholar
M.J. Corless, A.E. Frazho. Linear Systems and Control - An Operator Perspective. Marcel Dekker, New York, 2003.
Crum, M.M.. Associated Sturm-Liouville systems . Quart. J. Math. Oxford Ser. (2), 6 (1955), 121–127.CrossRefGoogle Scholar
G. Darboux. Lecons sur la Theorie Generale de Surface et les Applications Geometriques du Calcul Infinitesimal, II. Gauthiers-Villars, Paris, 1889.
Deift, P.A.. Applications of a commutation formula . Duke Math. J., 45 (1978), 267–310.CrossRefGoogle Scholar
L.D. Faddeev, L.A. Takhtajan. Hamiltonian methods in the theory of solitons. Springer Verlag, NY, 1986.
Fritzsche, B., Kirstein, B., Sakhnovich, A.L.. Completion problems and scattering problems for Dirac type differential equations with singularities . J. Math. Anal. Appl., 317 (2006), 510–525.CrossRefGoogle Scholar
B. Fritzsche, B. Kirstein, A.L. Sakhnovich. Semiseparable integral operators and explicit solution of an inverse problem for the skew-self-adjoint Dirac-type system. arXiv:0904.2357
Gesztesy, F.. A complete spectral characterization of the double commutation method . J. Funct. Anal., 117 (1993), No. 2, 401–446.CrossRefGoogle Scholar
F. Gesztesy, H. Holden.Soliton equations and their algebro-geometric solutions. Cambridge Studies in Advanced Mathematics, 79, Cambridge University Press, Cambridge, 2003.
Gesztesy, F., Simon, B., Teschl, G.. Spectral deformations of one-dimensional Schrödinger operators . J. Anal. Math., 70 (1996), 267-324.CrossRefGoogle Scholar
Gesztesy, F., Teschl, G.. On the double commutation method . Proc. Am. Math. Soc., 124 (1996), No. 6, 1831–1840.CrossRefGoogle Scholar
Gohberg, I., Kaashoek, M.A., Sakhnovich, A.L.. Canonical systems with rational spectral densities: explicit formulas and applications . Mathematische Nachr. 194 (1998), 93–125. CrossRefGoogle Scholar
Gohberg, I., Kaashoek, M.A., Sakhnovich, A.L.. Pseudocanonical systems with rational Weyl functions: explicit formulas and applications . J. Differ. Equations, 146 (1998), 375–398. CrossRefGoogle Scholar
Gohberg, I., Kaashoek, M.A., Sakhnovich, A.L.. Sturm-Liouville systems with rational Weyl functions: explicit formulas and applications . IEOT, 30 (1998), 338–377. Google Scholar
I. Gohberg, M.A. Kaashoek, A.L. Sakhnovich. Canonical systems on the full line with rational spectral densities: explicit formulas. In: Operator Theory: Adv. Appl., 117, M.G. Krein volume (2000), 127–139.
Gohberg, I., Kaashoek, M.A., Sakhnovich, A.L.. Bound states for canonical systems on the half and full line: explicit formulas . IEOT, 40 (2001), No. 3, 268–277.Google Scholar
Gohberg, I., Kaashoek, M.A., Sakhnovich, A.L.. Scattering problems for a canonical system with a pseudo-exponential potential . Asymptotic Analysis, 29 (2002), No. 1, 1–38.Google Scholar
C.H. Gu, H. Hu, Z. Zhou. Darboux transformations in integrable systems. Springer Verlag, 2005.
Jacobi, C.G.T.. Über eine neue Methode zur Integration der hyperelliptischen Differentialgleichungen und über die rationale Form ihrer vollständigen algebraischen Integralgleichungen . J. Reine Angew. Math., 32 (1846), 220–226.CrossRefGoogle Scholar
Jaworski, M., Kaup, D.. Direct and inverse scattering problem associated with the elliptic sinh-Gordon equation . Inverse Problems, 6 (1990), 543–556.CrossRefGoogle Scholar
Kaashoek, M.A., Sakhnovich, A.L.. Discrete skew self-adjoint canonical system and the isotropic Heisenberg magnet model . J. Funct. Anal., 228 (2005), 207–233.CrossRefGoogle Scholar
R.E. Kalman, P. Falb, M. Arbib. Topics in mathematical system theory. McGraw-Hill, NY, 1969.
Kasman, A., Gekhtman, M.. Solitons and almost-intertwining matrices . J. Math. Phys., 42 (2001), 3540–3551.CrossRefGoogle Scholar
Katsnelson, V.E.. Right and left joint system representation of a rational matrix function in general position . In: Operator Theory: Adv. Appl., 123 (2001), 337–400.Google Scholar
B.G. Konopelchenko, C. Rogers. Bäcklund and reciprocal transformations: gauge connections. In: Nonlinear equations in applied sciences (W.F. Ames, C. Rogers, eds.), Academic Press, San Diego, 1992, 317–362.
Kuznetsov, V.B., Salerno, M., Sklyanin, E.K.. Quantum Bäcklund transformation for the integrable DST model . J. Phys. A, 33 (2000), No. 1, 171–189.CrossRefGoogle Scholar
Levi, D., Ragnisco, O., Sym, A.. Dressing method vs. classical Darboux transformation . Nuovo Cimento B, 83 (1984), 34–41.CrossRefGoogle Scholar
P. Lancaster, L. Rodman,Algebraic Riccati equations. Clarendon Press, Oxford, 1995.
Liu, Q.P., Manas, M.. Vectorial Darboux transformations for the Kadomtsev-Petviashvili hierarchy . J. Nonlinear Sci., 9 (1999), No. 2, 213–232. CrossRefGoogle Scholar
V.A. Marchenko. Nonlinear equations and operator algebras. Reidel Publishing Co., Dordrecht, 1988.
Matveev, V.B.. Positons: slowly decaying soliton analogs . Teoret. Mat. Fiz., 131 (2002), No. 1, 44-61.Google Scholar
V.B. Matveev, M.A. Salle.Darboux transformations and solitons. Springer Verlag, Berlin, 1991.
Mennicken, R., Sakhnovich, A.L., Tretter, C.. Direct and inverse spectral problem for a system of differential equations depending rationally on the spectral parameter . Duke Math. J., 109 (2001), No. 3, 413–449.Google Scholar
R. Miura (ed.).Bäcklund Transformations. Lecture Notes in Math., 515, Springer-Verlag, Berlin, 1976.
Pohlmeyer, K.. Integrable Hamiltonian systems and interactions through quadratic constraints. Comm. Math. Phys., 46 (1976), No. 3, 207–221. CrossRefGoogle Scholar
C. Rogers, W.K. Schief.Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
Sattinger, D.S., Zurkowski, V.D.. Gauge theory of Bäcklund transformations II . Phys. D, 26 (1987), 225–250.CrossRefGoogle Scholar
Sakhnovich, A.L.. Nonlinear Schrödinger equation on a semi-axis and an inverse problem associated with it . Ukr. Math. J., 42 (1990), No. 3, 316-323. CrossRefGoogle Scholar
Sakhnovich, A.L.. The Goursat problem for the sine-Gordon equation and the inverse spectral problem . Russ. Math. Iz. VUZ, 36 (1992), No. 11, 42–52. Google Scholar
Sakhnovich, A.L.. Exact solutions of nonlinear equations and the method of operator identities . Lin. Alg. Appl., 182 (1993), 109–126. CrossRefGoogle Scholar
Sakhnovich, A.L.. Dressing procedure for solutions of nonlinear equations and the method of operator identities . Inverse Problems, 10 (1994), 699-710. CrossRefGoogle Scholar
Sakhnovich, A.L.. Iterated Darboux transform (the case of rational dependence on the spectral parameter) . Dokl. Natz. Akad. Nauk Ukrain., 7 (1995), 24–27. Google Scholar
Sakhnovich, A.L.. Iterated Bäcklund-Darboux transformation and transfer matrix-function (nonisospectral case) . Chaos, Solitons and Fractals, 7 (1996), 1251–1259. CrossRefGoogle Scholar
Sakhnovich, A.L.. Iterated Bäcklund-Darboux transform for canonical systems . J. Functional Anal., 144 (1997), 359–370. CrossRefGoogle Scholar
A.L. Sakhnovich. Inverse spectral problem related to the N -wave equation. In: Operator Theory: Adv. Appl., 117, M.G. Krein volume (2000), 323–338.
Sakhnovich, A.L.. Generalized Bäcklund-Darboux transformation: spectral properties and nonlinear equations . JMAA, 262 (2001), 274–306.Google Scholar
Sakhnovich, A.L.. Dirac type and canonical systems: spectral and Weyl-Titchmarsh fuctions, direct and inverse problems . Inverse Problems, 18 (2002), 331–348.CrossRefGoogle Scholar
Sakhnovich, A.L.. Dirac type system on the axis: explicit formulas for matrix potentials with singularities and soliton-positon interactions . Inverse Problems, 19 (2003), 845–854.CrossRefGoogle Scholar
Sakhnovich, A.L.. Non-Hermitian matrix Schrödinger equation: Bäcklund-Darboux transformation, Weyl functions, and 𝒫𝒯 symmetry . J. Phys. A, 36 (2003), 7789–7802.CrossRefGoogle Scholar
Sakhnovich, A.L.. Matrix Kadomtsev-Petviashvili equation: matrix identities and explicit non-singular solutions . J. Phys. A, 36 (2003), 5023–5033.CrossRefGoogle Scholar
Sakhnovich, A.L.. Second harmonic generation: Goursat problem on the semi-strip, Weyl functions and explicit solutions . Inverse Problems 21 (2005), No. 2, 703-716.CrossRefGoogle Scholar
Sakhnovich, A.L.. Non-self-adjoint Dirac-type systems and related nonlinear equations: wave functions, solutions, and explicit formulas . IEOT, 55 (2006), 127–143.Google Scholar
Sakhnovich, A.L.. Harmonic maps, Bäcklund-Darboux transformations and "line solution" analogues . J. Phys. A: Math. Gen., 39 (2006), 15379–15390. CrossRefGoogle Scholar
Sakhnovich, A.L.. Skew-self-adjoint discrete and continuous Dirac-type systems: inverse problems and Borg-Marchenko theorems . Inverse Problems, 22 (2006), 2083–2101.CrossRefGoogle Scholar
Sakhnovich, A.L.. Bäcklund-Darboux transformation for non-isospectral canonical system and Riemann-Hilbert problem . Symmetry Integrability Geom. Methods Appl., 3 (2007), 054.Google Scholar
Sakhnovich, A.L.. Discrete canonical system and non-Abelian Toda lattice: Bäcklund-Darboux transformation and Weyl functions . Math. Nachr., 280 (2007), No. 5-6, 1–23.CrossRefGoogle Scholar
Sakhnovich, A.L.. Weyl functions, inverse problem and special solutions for the system auxiliary to the nonlinear optics equation . Inverse Problems, 24 (2008), 025026.CrossRefGoogle Scholar
Sakhnovich, A.L.. Nonisospectral integrable nonlinear equations with external potentials and their GBDT solutions . J. Phys. A: Math. Theor., 41 (2008), 155204.CrossRefGoogle Scholar
Sakhnovich, A.L.. Weyl functions, inverse problem and special solutions for the system auxiliary to the nonlinear optics equation . Inverse Problems, 24 (2008), 025026.CrossRefGoogle Scholar
Sakhnovich, A.L., Zubelli, J.P.. Bundle bispectrality for matrix differential equations . IEOT, 41 (2001), 472–496.Google Scholar
Sakhnovich, L.A.. On the factorization of the transfer matrix function . Sov. Math. Dokl., 17 (1976), 203–207. Google Scholar
L.A. Sakhnovich.Spectral theory of canonical differential systems, method of operator identities. Operator Theory: Adv. Appl., 107, Birkhäuser Verlag, Basel-Boston, 1999.
Schiebold, C.. Explicit solution formulas for the matrix-KP . Glasg. Math. J., 51A (2009), 147–155.CrossRefGoogle Scholar
Terng, C.L., Uhlenbeck, K.. Bäcklund transformations and loop group actions . Commun. Pure Appl. Math., 53 (2000), 1–75.3.0.CO;2-U>CrossRefGoogle Scholar
Teschl, G.. Deforming the point spectra of one-dimensional Dirac operators . Proc. Amer. Math. Soc., 126 (1998), No. 10, 2873–2881.CrossRefGoogle Scholar
Wright, O.C., Forest, M.G.. On the Bäcklund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrödinger system . Physica D, 141 (2000), 104–116.CrossRefGoogle Scholar
Yagle, A.E., Levy, B.C.. The Schur algorithm and its applications . Acta Appl.Math., 3 (1985), 255–284. CrossRefGoogle Scholar
Zakharov, V.E., Manakov, S.V.. Theory of resonance interaction of wave packages in nonlinear medium . JETP, 69 (1975), No. 5, 1654–1673.Google Scholar
Zakharov, V.E., Mikhailov, A.V.. Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method (Russian). Soviet Phys. JETP, 74 (1978), No. 6, 1953–1973. Google Scholar
Zakharov, V.E., Mikhailov, A.V.. On the integrability of classical spinor models in two-dimensional space-time . Comm. Math. Phys., 74 (1980), 21–40.CrossRefGoogle Scholar
Zaharov, V.E., Shabat, A.B.. On soliton interaction in stable media . JETP, 64 (1973), 1627–1639.Google Scholar