Limits of soliton solutions
F. Gesztesy, W. Karwowski, and Z. Zhao
Source: Duke Math. J. Volume 68, Number 1
(1992), 101-150.
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References
[1] E. Brüning and F. Gesztesy, Continuity of wave and scattering operators with respect to interactions, J. Math. Phys. 24 (1983), no. 6, 1516–1528.
Mathematical Reviews (MathSciNet): MR85g:81179
Zentralblatt MATH: 0523.47008
Digital Object Identifier: doi:10.1063/1.525890
[2] R. Carmona and J. Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston Inc., Boston, MA, 1990.
Mathematical Reviews (MathSciNet): MR92k:47143
Zentralblatt MATH: 0717.60074
[3] W. E. Couch and R. J. Torrence, A particular $N$-soliton solution and scalar wave equations, J. Math. Phys. 20 (1979), no. 12, 2423–2426.
Mathematical Reviews (MathSciNet): MR80m:35064
Zentralblatt MATH: 0455.35106
Digital Object Identifier: doi:10.1063/1.524049
[4] W. Craig, The trace formula for Schrödinger operators on the line, Comm. Math. Phys. 126 (1989), no. 2, 379–407.
Mathematical Reviews (MathSciNet): MR90m:47063
Zentralblatt MATH: 0681.34026
Digital Object Identifier: doi:10.1007/BF02125131
Project Euclid: euclid.cmp/1104179857
[5] P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), no. 2, 267–310.
Mathematical Reviews (MathSciNet): MR81g:47001
Zentralblatt MATH: 0392.47013
Digital Object Identifier: doi:10.1215/S0012-7094-78-04516-7
Project Euclid: euclid.dmj/1077312819
[6] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121–251.
Mathematical Reviews (MathSciNet): MR80e:34011
Zentralblatt MATH: 0388.34005
Digital Object Identifier: doi:10.1002/cpa.3160320202
[7] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and nonlinear wave equations, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1982.
Mathematical Reviews (MathSciNet): MR84j:35142
Zentralblatt MATH: 0496.35001
[8] P. L. Duren, Theory of $H\spp$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York, 1970.
Mathematical Reviews (MathSciNet): MR42:3552
Zentralblatt MATH: 0215.20203
[9] F. J. Dyson, Old and new approaches to the inverse scattering problem, Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann eds. E. H. Lieb, B. Simon, and A. S. Wightman, Princeton Ser. Phys. Princeton Univ. Press, Princeton, 1976, pp. 151–167.
Zentralblatt MATH: 0343.47006
[10] N. Ercolani and H. P. McKean, Geometry of KdV. IV. Abel sums, Jacobi variety, and theta function in the scattering case, Invent. Math. 99 (1990), no. 3, 483–544.
Mathematical Reviews (MathSciNet): MR91f:58038
Zentralblatt MATH: 0705.58026
Digital Object Identifier: doi:10.1007/BF01234429
[11] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974), 97–133.
Mathematical Reviews (MathSciNet): MR49:898
Zentralblatt MATH: 0291.35012
Digital Object Identifier: doi:10.1002/cpa.3160270108
[12] I. M. Gel'fand and L. A. Dikii, Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations, Russ. Math. Surveys 30 (1975), no. 5, 77–113.
Zentralblatt MATH: 0334.58007
Mathematical Reviews (MathSciNet): MR508337
[13] F. Gesztesy, W. Schweiger, and B. Simon, Commutation methods applied to the mKdV-equation, Trans. Amer. Math. Soc. 324 (1991), no. 2, 465–525.
Mathematical Reviews (MathSciNet): MR92b:35132
Zentralblatt MATH: 0728.35106
Digital Object Identifier: doi:10.2307/2001730
[14] F. Gesztesy and H. Holden, , in preparation.
[15] D. J. Gilbert, On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 3-4, 213–229.
Mathematical Reviews (MathSciNet): MR90j:47061
Zentralblatt MATH: 0678.34024
[16] D. J. Gilbert and D. B. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987), no. 1, 30–56.
Mathematical Reviews (MathSciNet): MR89a:34033
Zentralblatt MATH: 0666.34023
Digital Object Identifier: doi:10.1016/0022-247X(87)90212-5
[17] H. Grosse, Quasiclassical estimates on moments of the energy levels, Acta Phys. Austriaca 52 (1980), no. 2, 89–105.
Mathematical Reviews (MathSciNet): MR82g:81017
[18] E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965.
Mathematical Reviews (MathSciNet): MR32:5826
Zentralblatt MATH: 0307.28001
[19] A. R. Its and V. B. Matveev, Schrödinger operators with the finite-gap spectrum and $N$-soliton solutions of the Korteweg-de Vries equation, Theoret. and Math. Phys. 23 (1975), 343–355.
[20] K. Iwasaki, Inverse problem for Sturm-Liouville and Hill equations, Ann. Mat. Pura Appl. (4) 149 (1987), 185–206.
Mathematical Reviews (MathSciNet): MR89d:34053
Zentralblatt MATH: 0641.34012
Digital Object Identifier: doi:10.1007/BF01773933
[21] I. S. Kac, On the multiplicity of the spectrum of a second-order differential operator, Sov. Math. Dokl. 3 (1962), 1035–1039.
Zentralblatt MATH: 0131.31201
[22] T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
Mathematical Reviews (MathSciNet): MR96a:47025
Zentralblatt MATH: 0836.47009
[23] I. Kay and H. E. Moses, Reflectionless transmission through dielectrics and scattering potentials, J. Appl. Phys. 27 (1956), 1503–1508.
Zentralblatt MATH: 0073.22202
Mathematical Reviews (MathSciNet): MR90367
[24] M. Klaus, On the bound state of Schrödinger operators in one dimension, Ann. Physics 108 (1977), no. 2, 288–300.
Mathematical Reviews (MathSciNet): MR58:20010
Zentralblatt MATH: 0427.47033
Digital Object Identifier: doi:10.1016/0003-4916(77)90015-X
[25] M. Klaus, On $-d\sp2/dx\sp2+V$ where $V$ has infinitely many “bumps”, Ann. Inst. H. Poincaré Sect. A (N.S.) 38 (1983), no. 1, 7–13.
Mathematical Reviews (MathSciNet): MR84m:34029
Zentralblatt MATH: 0527.47032
[26] S. Kotani and M. Krishna, Almost periodicity of some random potentials, J. Funct. Anal. 78 (1988), no. 2, 390–405.
Mathematical Reviews (MathSciNet): MR89i:60133
Zentralblatt MATH: 0644.60061
Digital Object Identifier: doi:10.1016/0022-1236(88)90125-5
[27] B. M. Levitan, On the closure of the set of finite-zone potentials, Math. USSR-Sb. 51 (1985), 67–89.
[28] E. H. Lieb and W. E. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann eds. E. H. Lieb, B. Simon, and A. S. Wightman, Princeton Ser. Phys., Princeton Univ. Press, Princeton, 1976, pp. 269–303.
Zentralblatt MATH: 0342.35044
[29] E. R. Lubenets, Estimates on the scattering data in a Schrödinger spectral problem on the line, Theoret. and Math. Phys. 79 (1989), 396–406.
Mathematical Reviews (MathSciNet): MR1000945
[30] D. Sh. Lundina, Compactness of sets of reflectionless potentials, Teor. Funktsiǐ Funktsional. Anal. i Prilozhen 44 (1985), 57–66, in Russian.
Zentralblatt MATH: 0583.47046
[31] F. Mantlik and A. Schneider, Note on the absolutely continuous spectrum of Sturm-Liouville operators, Math. Z. 205 (1990), no. 3, 491–498.
Mathematical Reviews (MathSciNet): MR92b:34103
Zentralblatt MATH: 0713.34030
Digital Object Identifier: doi:10.1007/BF02571258
[32] V. A. Marchenko, The Cauchy problem for the KdV equation with nondecreasing initial data, What is integrability? ed. V. E. Zakharov, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, pp. 273–318.
Mathematical Reviews (MathSciNet): MR92d:34156
Zentralblatt MATH: 0810.34090
[33] H. P. McKean, Theta functions, solitons, and singular curves, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977) ed. C. I. Byrnes, Lecture Notes in Pure and Appl. Math., vol. 48, Dekker, New York, 1979, pp. 237–254.
Mathematical Reviews (MathSciNet): MR82f:14028
Zentralblatt MATH: 0411.58023
[34] R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202–1204.
Mathematical Reviews (MathSciNet): MR40:6042a
Zentralblatt MATH: 0283.35018
Digital Object Identifier: doi:10.1063/1.1664700
[35] S. N. Naboko, Dense point spectra of Schrödinger and Dirac operators, Theoret. and Math. Phys. 68 (1986), 646–653.
Zentralblatt MATH: 0607.34023
Mathematical Reviews (MathSciNet): MR875178
[36] M. M. Nieto, Exact wave-function normalization constants for the $B_0\tanh z-U_0\cosh^-2z$ and Pöschl-Teller potentials, Phys. Rev. A. 17 (1978), no. 3, 1273–1283.
[37] G. Polya and G. Szegö, Problems and Theorems in Analysis, Volume II, Grundlehren Math. Wiss., vol. 216, Springer-Verlag, Berlin, 1976.
Zentralblatt MATH: 0311.00002
[38] M. Reed and B. Simon, Methods of modern mathematical physics. I, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980.
Mathematical Reviews (MathSciNet): MR85e:46002
Zentralblatt MATH: 0459.46001
[39] M. Reed and B. Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1979, Scattering Theory.
Mathematical Reviews (MathSciNet): MR80m:81085
Zentralblatt MATH: 0405.47007
[40] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.
Mathematical Reviews (MathSciNet): MR58:12429c
Zentralblatt MATH: 0401.47001
[41] U.-W. Schmincke, On Schrödinger's factorization method for Sturm-Liouville operators, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), no. 1-2, 67–84.
Mathematical Reviews (MathSciNet): MR80f:34025
Zentralblatt MATH: 0395.47022
[42] G. Segal and G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. (1985), no. 61, 5–65.
Mathematical Reviews (MathSciNet): MR87b:58039
Zentralblatt MATH: 0592.35112
Digital Object Identifier: doi:10.1007/BF02698802
[43] B. Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Physics 97 (1976), no. 2, 279–288.
Mathematical Reviews (MathSciNet): MR53:8646
Zentralblatt MATH: 0325.35029
Digital Object Identifier: doi:10.1016/0003-4916(76)90038-5
[44] B. Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge, 1979.
Mathematical Reviews (MathSciNet): MR80k:47048
Zentralblatt MATH: 0423.47001
[45] S. Venakides, The generation of modulated wavetrains in the solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 38 (1985), no. 6, 883–909.
Mathematical Reviews (MathSciNet): MR87c:35148
Zentralblatt MATH: 0657.35110
Digital Object Identifier: doi:10.1002/cpa.3160380616
[46] V. E. Zakharov and L. D. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl. 5 (1971), 280–287.
Zentralblatt MATH: 0257.35074
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