## Duke Mathematical Journal

### Limits of soliton solutions

#### Article information

Source
Duke Math. J. Volume 68, Number 1 (1992), 101-150.

Dates
First available: 20 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077293866

Mathematical Reviews number (MathSciNet)
MR1185820

Zentralblatt MATH identifier
0811.35122

Digital Object Identifier
doi:10.1215/S0012-7094-92-06805-0

#### Citation

Gesztesy, F.; Karwowski, W.; Zhao, Z. Limits of soliton solutions. Duke Mathematical Journal 68 (1992), no. 1, 101--150. doi:10.1215/S0012-7094-92-06805-0. http://projecteuclid.org/euclid.dmj/1077293866.

#### 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.
• [2] R. Carmona and J. Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston Inc., Boston, MA, 1990.
• [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.
• [4] W. Craig, The trace formula for Schrödinger operators on the line, Comm. Math. Phys. 126 (1989), no. 2, 379–407.
• [5] P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), no. 2, 267–310.
• [6] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121–251.
• [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.
• [8] P. L. Duren, Theory of $H\spp$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York, 1970.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [17] H. Grosse, Quasiclassical estimates on moments of the energy levels, Acta Phys. Austriaca 52 (1980), no. 2, 89–105.
• [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.
• [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.
• [21] I. S. Kac, On the multiplicity of the spectrum of a second-order differential operator, Sov. Math. Dokl. 3 (1962), 1035–1039.
• [22] T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
• [23] I. Kay and H. E. Moses, Reflectionless transmission through dielectrics and scattering potentials, J. Appl. Phys. 27 (1956), 1503–1508.
• [24] M. Klaus, On the bound state of Schrödinger operators in one dimension, Ann. Physics 108 (1977), no. 2, 288–300.
• [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.
• [26] S. Kotani and M. Krishna, Almost periodicity of some random potentials, J. Funct. Anal. 78 (1988), no. 2, 390–405.
• [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.
• [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.
• [30] D. Sh. Lundina, Compactness of sets of reflectionless potentials, Teor. Funktsiǐ Funktsional. Anal. i Prilozhen 44 (1985), 57–66, in Russian.
• [31] F. Mantlik and A. Schneider, Note on the absolutely continuous spectrum of Sturm-Liouville operators, Math. Z. 205 (1990), no. 3, 491–498.
• [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.
• [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.
• [34] R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202–1204.
• [35] S. N. Naboko, Dense point spectra of Schrödinger and Dirac operators, Theoret. and Math. Phys. 68 (1986), 646–653.
• [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.
• [38] M. Reed and B. Simon, Methods of modern mathematical physics. I, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980.
• [39] M. Reed and B. Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1979, Scattering Theory.
• [40] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.
• [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.
• [42] G. Segal and G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. (1985), no. 61, 5–65.
• [43] B. Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Physics 97 (1976), no. 2, 279–288.
• [44] B. Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge, 1979.
• [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.
• [46] V. E. Zakharov and L. D. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl. 5 (1971), 280–287.