Free Access
Math. Model. Nat. Phenom.
Volume 5, Number 4, 2010
Spectral problems. Issue dedicated to the memory of M. Birman
Page(s) 340 - 389
Published online 12 May 2010
  1. M.J. Ablowitz, S. Chakravarty, A.D. Trubatch, J. Villarroel. 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. [CrossRef] [MathSciNet]
  2. M.J. Ablowitz, R. Haberman. Resonantly coupled nonlinear evolution equations. J. Math. Phys., 16 (1975), 2301–2305. [CrossRef]
  3. M. Adler, P. van Moerbeke. Birkhoff strata, Bäcklund transformations, and regularization of isospectral operators. Adv. Math., 108 (1994), No. 1, 140–204. [CrossRef] [MathSciNet]
  4. S. Albeverio, R. Hryniv, Ya. Mykytyuk. Reconstruction of radial Dirac operators.J. Math. Phys. 48 (2007), No. 4, 043501, 14 pp.
  5. D. Alpay, I. Gohberg. Inverse spectral problem for differential operators with rational scattering matrix functions. J. Diff. Eqs., 118 (1995), 1–19. [CrossRef]
  6. D. Alpay, I. Gohberg, M.A. Kaashoek, A.L. Sakhnovich. Direct and inverse scattering problem for canonical systems with a strictly pseudo-exponential potential. Math. Nachr., 215 (2000), 5–31. [CrossRef] [MathSciNet]
  7. A.V. Bäcklund. Zur Theorie der partiellen Differential gleichungen erster Ordnung. Math. Ann., 17 (1880), 285–328. [CrossRef] [MathSciNet]
  8. H. Bart, I. Gohberg, M.A. Kaashoek.Minimal factorization of matrix and operator functions. Operator Theory: Adv. Appl., 1, Birkhäuser Verlag, Basel, 1979.
  9. R. Beals, R.R. Coifman. Scattering and inverse scattering for first-order systems: II. Inverse Probl., 3 (1987), 577–593. [CrossRef]
  10. A.B. Borisov, V.V. Kiseliev. Inverse problem for an elliptic sine-Gordon equation with an asymptotic behaviour of the cnoidal-wave type. Inverse Probl., 5 (1989), 959–982. [CrossRef]
  11. A. Boutet de Monvel, V. Marchenko. Generalization of the Darboux transform. Matematicheskaya fizika, analiz, geometriya, 1 (1994), 479–504. [MathSciNet]
  12. B. Carl, C. Schiebold. Nonlinear equations in soliton physics and operator ideals. Nonlinearity, 12 (1999), 333–364. [CrossRef] [MathSciNet]
  13. R.C. Cascaval, F. Gesztesy, H. Holden, Yu. Latushkin. Spectral analysis of Darboux transformations for the focusing NLS hierarchy. J. Anal. Math., 93 (2004), 139–197. [CrossRef] [MathSciNet]
  14. D.V. Chudnovsky, G.V. Chudnovsky. Bäcklund transformation as a method of decomposition and reproduction of two-dimensional nonlinear systems. Phys. Lett. A, 87 (1982), No. 7, 325–329. [CrossRef]
  15. J. Cieslinski. An effective method to compute N-fold Darboux matrix and N-soliton surfaces. J. Math. Phys., 32 (1991), 2395–2399. [CrossRef] [MathSciNet]
  16. S. Clark, F. Gesztesy. On self-adjoint andJ-self-adjoint Dirac-type operators: a case study. Contemporary Mathematics, 412 (2006), 103–140.
  17. M.J. Corless, A.E. Frazho. Linear Systems and Control - An Operator Perspective. Marcel Dekker, New York, 2003.
  18. M.M. Crum. Associated Sturm-Liouville systems. Quart. J. Math. Oxford Ser. (2), 6 (1955), 121–127. [CrossRef] [MathSciNet]
  19. G. Darboux. Lecons sur la Theorie Generale de Surface et les Applications Geometriques du Calcul Infinitesimal, II. Gauthiers-Villars, Paris, 1889.
  20. P.A. Deift. Applications of a commutation formula. Duke Math. J., 45 (1978), 267–310. [CrossRef] [MathSciNet]
  21. L.D. Faddeev, L.A. Takhtajan. Hamiltonian methods in the theory of solitons. Springer Verlag, NY, 1986.
  22. B. Fritzsche, B. Kirstein, A.L. Sakhnovich. Completion problems and scattering problems for Dirac type differential equations with singularities. J. Math. Anal. Appl., 317 (2006), 510–525. [CrossRef] [MathSciNet]
  23. 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
  24. F. Gesztesy. A complete spectral characterization of the double commutation method. J. Funct. Anal., 117 (1993), No. 2, 401–446. [CrossRef] [MathSciNet]
  25. F. Gesztesy, H. Holden.Soliton equations and their algebro-geometric solutions. Cambridge Studies in Advanced Mathematics, 79, Cambridge University Press, Cambridge, 2003.
  26. F. Gesztesy, B. Simon, G. Teschl. Spectral deformations of one-dimensional Schrödinger operators. J. Anal. Math., 70 (1996), 267-324. [CrossRef] [MathSciNet]
  27. F. Gesztesy, G. Teschl. On the double commutation method. Proc. Am. Math. Soc., 124 (1996), No. 6, 1831–1840. [CrossRef]
  28. I. Gohberg, M.A. Kaashoek, A.L. Sakhnovich. Canonical systems with rational spectral densities: explicit formulas and applications. Mathematische Nachr. 194 (1998), 93–125. [CrossRef]
  29. I. Gohberg, M.A. Kaashoek, A.L. Sakhnovich. Pseudocanonical systems with rational Weyl functions: explicit formulas and applications. J. Differ. Equations, 146 (1998), 375–398. [CrossRef]
  30. I. Gohberg, M.A. Kaashoek, A.L. Sakhnovich. Sturm-Liouville systems with rational Weyl functions: explicit formulas and applications. IEOT, 30 (1998), 338–377.
  31. 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.
  32. I. Gohberg, M.A. Kaashoek, A.L. Sakhnovich. Bound states for canonical systems on the half and full line: explicit formulas. IEOT, 40 (2001), No. 3, 268–277.
  33. I. Gohberg, M.A. Kaashoek, A.L. Sakhnovich. Scattering problems for a canonical system with a pseudo-exponential potential. Asymptotic Analysis, 29 (2002), No. 1, 1–38. [MathSciNet]
  34. C.H. Gu, H. Hu, Z. Zhou. Darboux transformations in integrable systems. Springer Verlag, 2005.
  35. C.G.T. Jacobi. 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.
  36. M. Jaworski, D. Kaup. Direct and inverse scattering problem associated with the elliptic sinh-Gordon equation. Inverse Problems, 6 (1990), 543–556. [CrossRef] [MathSciNet]
  37. M.A. Kaashoek, A.L. Sakhnovich. Discrete skew self-adjoint canonical system and the isotropic Heisenberg magnet model. J. Funct. Anal., 228 (2005), 207–233. [CrossRef] [MathSciNet]
  38. R.E. Kalman, P. Falb, M. Arbib. Topics in mathematical system theory. McGraw-Hill, NY, 1969.
  39. A. Kasman, M. Gekhtman. Solitons and almost-intertwining matrices. J. Math. Phys., 42 (2001), 3540–3551. [CrossRef] [MathSciNet]
  40. V.E. Katsnelson. Right and left joint system representation of a rational matrix function in general position. In: Operator Theory: Adv. Appl., 123 (2001), 337–400.
  41. 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.
  42. V.B. Kuznetsov, M. Salerno, E.K. Sklyanin. Quantum Bäcklund transformation for the integrable DST model. J. Phys. A, 33 (2000), No. 1, 171–189. [CrossRef] [MathSciNet]
  43. D. Levi, O. Ragnisco, A. Sym. Dressing method vs. classical Darboux transformation. Nuovo Cimento B, 83 (1984), 34–41.
  44. P. Lancaster, L. Rodman,Algebraic Riccati equations. Clarendon Press, Oxford, 1995.
  45. Q.P. Liu, M. Manas. Vectorial Darboux transformations for the Kadomtsev-Petviashvili hierarchy. J. Nonlinear Sci., 9 (1999), No. 2, 213–232. [CrossRef] [MathSciNet]
  46. V.A. Marchenko. Nonlinear equations and operator algebras. Reidel Publishing Co., Dordrecht, 1988.
  47. V.B. Matveev. Positons: slowly decaying soliton analogs. Teoret. Mat. Fiz., 131 (2002), No. 1, 44-61.
  48. V.B. Matveev, M.A. Salle.Darboux transformations and solitons. Springer Verlag, Berlin, 1991.
  49. R. Mennicken, A.L. Sakhnovich, C. Tretter. 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. [CrossRef] [MathSciNet]
  50. R. Miura (ed.).Bäcklund Transformations. Lecture Notes in Math., 515, Springer-Verlag, Berlin, 1976.
  51. K. Pohlmeyer. Integrable Hamiltonian systems and interactions through quadratic constraints. Comm. Math. Phys., 46 (1976), No. 3, 207–221. [CrossRef] [MathSciNet]
  52. 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.
  53. D.S. Sattinger, V.D. Zurkowski. Gauge theory of Bäcklund transformations II. Phys. D, 26 (1987), 225–250. [CrossRef] [MathSciNet]
  54. A.L. Sakhnovich. Nonlinear Schrödinger equation on a semi-axis and an inverse problem associated with it. Ukr. Math. J., 42 (1990), No. 3, 316-323. [CrossRef]
  55. A.L. Sakhnovich. The Goursat problem for the sine-Gordon equation and the inverse spectral problem. Russ. Math. Iz. VUZ, 36 (1992), No. 11, 42–52.
  56. A.L. Sakhnovich. Exact solutions of nonlinear equations and the method of operator identities. Lin. Alg. Appl., 182 (1993), 109–126. [CrossRef]
  57. A.L. Sakhnovich. Dressing procedure for solutions of nonlinear equations and the method of operator identities. Inverse Problems, 10 (1994), 699-710. [CrossRef] [MathSciNet]
  58. A.L. Sakhnovich. Iterated Darboux transform (the case of rational dependence on the spectral parameter). Dokl. Natz. Akad. Nauk Ukrain., 7 (1995), 24–27.
  59. A.L. Sakhnovich. Iterated Bäcklund-Darboux transformation and transfer matrix-function (nonisospectral case). Chaos, Solitons and Fractals, 7 (1996), 1251–1259. [CrossRef] [MathSciNet]
  60. A.L. Sakhnovich. Iterated Bäcklund-Darboux transform for canonical systems. J. Functional Anal., 144 (1997), 359–370. [CrossRef]
  61. 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.
  62. A.L. Sakhnovich. Generalized Bäcklund-Darboux transformation: spectral properties and nonlinear equations. JMAA, 262 (2001), 274–306.
  63. A.L. Sakhnovich. Dirac type and canonical systems: spectral and Weyl-Titchmarsh fuctions, direct and inverse problems. Inverse Problems, 18 (2002), 331–348. [CrossRef] [MathSciNet]
  64. 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. [CrossRef] [MathSciNet]
  65. A.L. Sakhnovich. Non-Hermitian matrix Schrödinger equation: Bäcklund-Darboux transformation, Weyl functions, and 𝒫𝒯 symmetry. J. Phys. A, 36 (2003), 7789–7802. [CrossRef] [MathSciNet]
  66. A.L. Sakhnovich. Matrix Kadomtsev-Petviashvili equation: matrix identities and explicit non-singular solutions. J. Phys. A, 36 (2003), 5023–5033. [CrossRef] [MathSciNet]
  67. A.L. Sakhnovich. Second harmonic generation: Goursat problem on the semi-strip, Weyl functions and explicit solutions. Inverse Problems 21 (2005), No. 2, 703-716. [CrossRef] [MathSciNet]
  68. A.L. Sakhnovich. Non-self-adjoint Dirac-type systems and related nonlinear equations: wave functions, solutions, and explicit formulas. IEOT, 55 (2006), 127–143.
  69. A.L. Sakhnovich. Harmonic maps, Bäcklund-Darboux transformations and "line solution" analogues. J. Phys. A: Math. Gen., 39 (2006), 15379–15390. [CrossRef]
  70. A.L. Sakhnovich. Skew-self-adjoint discrete and continuous Dirac-type systems: inverse problems and Borg-Marchenko theorems. Inverse Problems, 22 (2006), 2083–2101. [CrossRef] [MathSciNet]
  71. A.L. Sakhnovich. Bäcklund-Darboux transformation for non-isospectral canonical system and Riemann-Hilbert problem. Symmetry Integrability Geom. Methods Appl., 3 (2007), 054.
  72. A.L. Sakhnovich. Discrete canonical system and non-Abelian Toda lattice: Bäcklund-Darboux transformation and Weyl functions. Math. Nachr., 280 (2007), No. 5-6, 1–23.
  73. A.L. Sakhnovich. Weyl functions, inverse problem and special solutions for the system auxiliary to the nonlinear optics equation. Inverse Problems, 24 (2008), 025026. [CrossRef] [MathSciNet]
  74. A.L. Sakhnovich. Nonisospectral integrable nonlinear equations with external potentials and their GBDT solutions. J. Phys. A: Math. Theor., 41 (2008), 155204. [CrossRef]
  75. A.L. Sakhnovich. Weyl functions, inverse problem and special solutions for the system auxiliary to the nonlinear optics equation. Inverse Problems, 24 (2008), 025026. [CrossRef] [MathSciNet]
  76. A.L. Sakhnovich, J.P. Zubelli. Bundle bispectrality for matrix differential equations. IEOT, 41 (2001), 472–496.
  77. L.A. Sakhnovich. On the factorization of the transfer matrix function. Sov. Math. Dokl., 17 (1976), 203–207.
  78. L.A. Sakhnovich.Spectral theory of canonical differential systems, method of operator identities. Operator Theory: Adv. Appl., 107, Birkhäuser Verlag, Basel-Boston, 1999.
  79. C. Schiebold. Explicit solution formulas for the matrix-KP. Glasg. Math. J., 51A (2009), 147–155. [CrossRef]
  80. C.L. Terng, K. Uhlenbeck. Bäcklund transformations and loop group actions. Commun. Pure Appl. Math., 53 (2000), 1–75. [CrossRef]
  81. G. Teschl. Deforming the point spectra of one-dimensional Dirac operators. Proc. Amer. Math. Soc., 126 (1998), No. 10, 2873–2881. [CrossRef] [MathSciNet]
  82. O.C. Wright, M.G. Forest. On the Bäcklund-gauge transformation and homoclinic orbits of a coupled nonlinear Schrödinger system. Physica D, 141 (2000), 104–116. [CrossRef] [MathSciNet]
  83. A.E. Yagle, B.C. Levy. The Schur algorithm and its applications. Acta Appl.Math., 3 (1985), 255–284. [CrossRef] [MathSciNet]
  84. V.E. Zakharov, S.V. Manakov. Theory of resonance interaction of wave packages in nonlinear medium. JETP, 69 (1975), No. 5, 1654–1673.
  85. V.E. Zakharov, A.V. Mikhailov. 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.
  86. V.E. Zakharov, A.V. Mikhailov. On the integrability of classical spinor models in two-dimensional space-time. Comm. Math. Phys., 74 (1980), 21–40. [CrossRef] [MathSciNet]
  87. V.E. Zaharov, A.B. Shabat. On soliton interaction in stable media. JETP, 64 (1973), 1627–1639.

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.