|
New types of soliton solutions
Authors:
F. Gesztesy, W. Karwowski and Z. Zhao
Journal:
Bull. Amer. Math. Soc. 27 (1992), 266-272
MSC (2000):
Primary 35Q51; Secondary 35Q53
MathSciNet review:
1152159
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
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.
- [1]
P.
A. Deift, Applications of a commutation formula, Duke Math. J.
45 (1978), no. 2, 267–310. MR 495676
(81g:47001)
- [2]
P.
Deift and E.
Trubowitz, Inverse scattering on the line, Comm. Pure Appl.
Math. 32 (1979), no. 2, 121–251. MR 512420
(80e:34011), http://dx.doi.org/10.1002/cpa.3160320202
- [3]
Peter
L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and
Applied Mathematics, Vol. 38, Academic Press, New York, 1970. MR 0268655
(42 #3552)
- [4]
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
0336122 (49 #898)
- [5]
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
(58 #22746)
- [6]
F. Gesztesy and H. Holden, in preparation.
- [7]
F.
Gesztesy, W.
Karwowski, and Z.
Zhao, Limits of soliton solutions, Duke Math. J.
68 (1992), no. 1, 101–150. MR 1185820
(94b:35242), http://dx.doi.org/10.1215/S0012-7094-92-06805-0
- [8]
H.
Grosse, Quasiclassical estimates on moments of the energy
levels, Acta Phys. Austriaca 52 (1980), no. 2,
89–105. MR
584458 (82g:81017)
- [9]
Edwin
Hewitt and Karl
Stromberg, Real and abstract analysis, Springer-Verlag, New
York, 1975. A modern treatment of the theory of functions of a real
variable; Third printing; Graduate Texts in Mathematics, No. 25. MR 0367121
(51 #3363)
- [10]
E. H. Lieb and W. E. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities (E. H. Lieb, B. Simon, and A. S. Wightman, eds.), Studies in Mathematical Physics, Princeton Univ. Press, Princeton, NJ, 1976, pp. 269-303.
- [11]
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
(92b:34103), http://dx.doi.org/10.1007/BF02571258
- [12]
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
(80f:34025), http://dx.doi.org/10.1017/S0308210500010143
- [13]
V. E. Zakharov and L. D. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl. 5 (1971), 280-287.
- [1]
- P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), 267-310. MR 495676 (81g:47001)
- [2]
- P. Deift and E. Trubowitz, Inverse scattering on the line, Commun. Pure Appl. Math. 32 (1979), 121-251. MR 512420 (80e:34011)
- [3]
- P. L. Duren, Theory of
 Spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
- [4]
- C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Korteweg-de Vries equation and generalizations, VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974), 97-133. MR 0336122 (49:898)
- [5]
- I. M. Gel'fand and L. A. Dikii, Asymptotic behavior of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations, Russian Math. Surveys 30:5 (1975), 77-113. MR 0508337 (58:22746)
- [6]
- F. Gesztesy and H. Holden, in preparation.
- [7]
- F. Gesztesy, W. Karwowski and Z. Zhao, Limits of soliton solutions, Duke Math. J. (to appear). MR 1185820 (94b:35242)
- [8]
- H. Grosse, Quasiclassical estimates on moments of the energy levels, Acta Phys. Austriaca 52 (1980), 89-105. MR 584458 (82g:81017)
- [9]
- E. Hewitt and K. Stromberg, Real and abstract analysis, Springer, New York, 1965. MR 0367121 (51:3363)
- [10]
- E. H. Lieb and W. E. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities (E. H. Lieb, B. Simon, and A. S. Wightman, eds.), Studies in Mathematical Physics, Princeton Univ. Press, Princeton, NJ, 1976, pp. 269-303.
- [11]
- F. Mantlik and A. Schneider, Note on the absolutely continuous spectrum of Sturm-Liouville operators, Math. Z. 205 (1990), 491-498. MR 1082870 (92b:34103)
- [12]
- U.-W. Schmincke, On Schrödinger's factorization method for Sturm-Liouville operators, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), 67-84. MR 529570 (80f:34025)
- [13]
- V. E. Zakharov and L. D. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl. 5 (1971), 280-287.
Similar Articles
Retrieve articles in Bulletin of the American Mathematical Society
with MSC (2000):
35Q51,
35Q53
Retrieve articles in all journals
with MSC (2000):
35Q51,
35Q53
Additional Information
DOI:
http://dx.doi.org/10.1090/S0273-0979-1992-00309-9
PII:
S 0273-0979(1992)00309-9
Article copyright:
© Copyright 1992 American Mathematical Society
|
