New types of soliton solutions

Gesztesy, F.; Karwowski, W.; Zhao, Z.

doi:10.1090/S0273-0979-1992-00309-9

New types of soliton solutions
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by F. Gesztesy, W. Karwowski and Z. Zhao PDF

Bull. Amer. Math. Soc. 27 (1992), 266-272 Request permission

Abstract:

We announce a detailed investigation of limits of N-soliton solutions of the Korteweg-deVries (KdV) equation as N tends to infinity. Our main results provide new classes of KdV-solutions including in particular new types of soliton-like (reflectionless) solutions. As a byproduct we solve an inverse spectral problem for one-dimensional Schrödinger operators and explicitly construct smooth and real-valued potentials that yield a purely absolutely continuous spectrum on the nonnegative real axis and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of the negative real axis.

References

P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), no. 2, 267–310. MR 495676, DOI 10.1215/S0012-7094-78-04516-7
P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121–251. MR 512420, DOI 10.1002/cpa.3160320202
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Clifford S. Gardner, John M. Greene, Martin D. Kruskal, and Robert M. Miura, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974), 97–133. MR 336122, DOI 10.1002/cpa.3160270108
I. M. Gel′fand and L. A. Dikiĭ, Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations, Uspehi Mat. Nauk 30 (1975), no. 5(185), 67–100 (Russian). MR 0508337
F. Gesztesy, W. Karwowski, and Z. Zhao, Limits of soliton solutions, Duke Math. J. 68 (1992), no. 1, 101–150. MR 1185820, DOI 10.1215/S0012-7094-92-06805-0
H. Grosse, Quasiclassical estimates on moments of the energy levels, Acta Phys. Austriaca 52 (1980), no. 2, 89–105. MR 584458
Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121

Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities

Frank Mantlik and Albert Schneider, Note on the absolutely continuous spectrum of Sturm-Liouville operators, Math. Z. 205 (1990), no. 3, 491–498. MR 1082870, DOI 10.1007/BF02571258
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. MR 529570, DOI 10.1017/S0308210500010143

Korteweg-de Vries equation: A completely integrable Hamiltonian system

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