|
This article is cited in 70 scientific papers (total in 70 papers)
Schrödinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg–de Vries equation
A. R. Its, V. B. Matveev
Abstract:
Explicit description of periodic potentials for which the corresponding Schrodinger
operator $N$ possesses only the finite number of energy gaps is obtained. Using this result
the solution of the Korteveg–de Vries equation with the “finite-gap” initial condition
is expressed, by means of the $N$-dimensional $\Theta$-function, $N$ being the number of
the nondegenerate energy gaps. The following characteristic property of the $N$-gap
periodic potentials and the $N$-soliton decreasing potentials is discovered: the existence
of two solutions $\psi_1(x,\lambda), \psi_2(x,\lambda)$ of the Schrodinger equation, for which the product $\psi_1,\psi_2$ is the polynomial $P$ ($\operatorname{deg}P=N$. $N$ is the number of gaps or the number of bound states of $H$) from the spectral parameter $\lambda$.
Full text:
PDF file (1052 kB)
References:
PDF file
HTML file
English version:
Theoretical and Mathematical Physics, 1975, 23:1, 343–355
Bibliographic databases:
Received: 09.07.1974
Citation:
A. R. Its, V. B. Matveev, “Schrödinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg–de Vries equation”, TMF, 23:1 (1975), 51–68; Theoret. and Math. Phys., 23:1 (1975), 343–355
Citation in format AMSBIB
\Bibitem{ItsMat75}
\by A.~R.~Its, V.~B.~Matveev
\paper Schr\"odinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg--de~Vries equation
\jour TMF
\yr 1975
\vol 23
\issue 1
\pages 51--68
\mathnet{http://mi.mathnet.ru/tmf3750}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=479120}
\transl
\jour Theoret. and Math. Phys.
\yr 1975
\vol 23
\issue 1
\pages 343--355
\crossref{https://doi.org/10.1007/BF01038218}
Linking options:
http://mi.mathnet.ru/eng/tmf3750 http://mi.mathnet.ru/eng/tmf/v23/i1/p51
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
This publication is cited in the following articles:
-
V. A. Marchenko, I. V. Ostrovskii, “A characterization of the spectrum of Hill's operator”, Math. USSR-Sb., 26:4 (1975), 493–554
-
A. R. Its, V. B. Matveev, “vHill's operator with finitely many gaps”, Funct. Anal. Appl., 9:1 (1975), 65–66
-
B. A. Dubrovin, V. B. Matveev, S. P. Novikov, “Non-linear equations of Korteweg–de Vries type, finite-zone linear
operators, and Abelian varieties”, Russian Math. Surveys, 31:1 (1976), 59–146
-
E. Ya. Khruslov, “Asymptotics of the solution of the Cauchy problem for the Korteweg–de Vries equation with initial data of step type”, Math. USSR-Sb., 28:2 (1976), 229–248
-
I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry”, Funct. Anal. Appl., 11:1 (1977), 12–26
-
I. M. Krichever, “Methods of algebraic geometry in the theory of non-linear equations”, Russian Math. Surveys, 32:6 (1977), 185–213
-
L. A. Bordag, V. B. Matveev, “Self-similar solutions of the Korteweg–de Vries equation and potentials with a trivial $S$-matrix”, Theoret. and Math. Phys., 34:3 (1978), 272–275
-
I. M. Gel'fand, L. A. Dikii, “Integrable nonlinear equations and the Liouville theorem”, Funct. Anal. Appl., 13:1 (1979), 6–15
-
E. D. Belokolos, “Peierls-Fröhlich problem and potentials with finite number of gaps. II”, Theoret. and Math. Phys., 48:1 (1981), 604–610
-
B. A. Dubrovin, “Theta functions and non-linear equations”, Russian Math. Surveys, 36:2 (1981), 11–92
-
B. M. Levitan, “Almost periodicity of infinite-zone potentials”, Math. USSR-Izv., 18:2 (1982), 249–273
-
A. K. Prikarpatskii, “Almost periodic solutions of a modified nonlinear Schrödinger equation”, Theoret. and Math. Phys., 47:3 (1981), 487–493
-
V. P. Maslov, “Non-standard characteristics in asymptotic problems”, Russian Math. Surveys, 38:6 (1983), 1–42
-
M. V. Babich, A. I. Bobenko, V. B. Matveev, “Solutions of nonlinear equations integrable in Jacobi theta functions by the method of the inverse problem, and symmetries of algebraic curves”, Math. USSR-Izv., 26:3 (1986), 479–496
-
E. D. Belokolos, A. I. Bobenko, V. B. Matveev, V. Z. Ènol'skii, “Algebraic-geometric principles of superposition of finite-zone solutions of integrable non-linear equations”, Russian Math. Surveys, 41:2 (1986), 1–49
-
N. E. Firsova, “The direct and inverse scattering problems for the one-dimensional perturbed Hill operator”, Math. USSR-Sb., 58:2 (1987), 351–388
-
A. I. Bobenko, D. A. Kubenskii, “Qualitative analysis and calculations of finite-gap solutions of the Korteweg–de Vries equation. Automorphic approach”, Theoret. and Math. Phys., 72:3 (1987), 929–935
-
Yu. M. Vorob'ev, S. Yu. Dobrokhotov, “Completeness of the system of eigenfunctions of a nonelliptic operator on the torus, generated by a Hill operator with a finite-zone potential”, Funct. Anal. Appl., 22:2 (1988), 137–139
-
I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications”, Russian Math. Surveys, 44:2 (1989), 145–225
-
E. D. Belokolos, V. Z. Ènol'skii, “Isospectral deformations of elliptic potentials”, Russian Math. Surveys, 44:5 (1989), 191–193
-
B. A. Dubrovin, S. P. Novikov, “Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory”, Russian Math. Surveys, 44:6 (1989), 35–124
-
B. M. Levitan, A. B. Khasanov, “Estimation of the Cauchy function for finite-zone nonperiodic potentials”, Funct. Anal. Appl., 26:2 (1992), 91–98
-
E. L. Korotyaev, N. E. Firsova, “Diffusion in layered media at large time”, Theoret. and Math. Phys., 98:1 (1994), 72–99
-
A. O. Smirnov, “Solutions of the KdV equation elliptic in $t$”, Theoret. and Math. Phys., 100:2 (1994), 937–947
-
A. O. Smirnov, “Two-gap elliptic solutions to integrable nonlinear equations”, Math. Notes, 58:1 (1995), 735–743
-
S. P. Novikov, I. A. Dynnikov, “Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds”, Russian Math. Surveys, 52:5 (1997), 1057–1116
-
Gesztesy, F, “Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies - An analytic approach”, Bulletin of the American Mathematical Society, 35:4 (1998), 271
-
Dickson, R, “Algebro-geometric solutions of the Boussinesq hierarchy”, Reviews in Mathematical Physics, 11:7 (1999), 823
-
P. G. Grinevich, “Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity”, Russian Math. Surveys, 55:6 (2000), 1015–1083
-
A. Treibich, “Hyperelliptic tangential covers and finite-gap potentials”, Russian Math. Surveys, 56:6 (2001), 1107–1151
-
V. B. Matveev, “Positons: Slowly Decreasing Analogues of Solitons”, Theoret. and Math. Phys., 131:1 (2002), 483–497
-
G. A. El, “The Infinite-Genus Limit of the Whitham Equations”, Theoret. and Math. Phys., 137:2 (2003), 1505–1514
-
E. D. Belokolos, V. Z. Ènol'skii, M. Salerno, “Wannier Functions for Quasiperiodic Finite-Gap Potentials”, Theoret. and Math. Phys., 144:2 (2005), 1081–1099
-
Beals, R, “Periodic peakons and Calogero-Francoise flows”, Journal of the Institute of Mathematics of Jussieu, 4:1 (2005), 1
-
Chalykh, O, “Algebro-geometric Schrodinger operators in many dimensions”, Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 366:1867 (2008), 947
-
Matveev, VB, “30 years of finite-gap integration theory”, Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 366:1867 (2008), 837
-
E. Yu. Bunkova, V. M. Buchstaber, “Heat Equations and Families of Two-Dimensional Sigma Functions”, Proc. Steklov Inst. Math., 266 (2009), 1–28
-
A. B. Khasanov, A. B. Yakhshimuratov, “The Korteweg–de Vries equation with a self-consistent source in the class of periodic functions”, Theoret. and Math. Phys., 164:2 (2010), 1008–1015
-
J. Harnad, V. Z. Enolski, “Schur function expansions of KP $\tau$-functions associated to algebraic curves”, Russian Math. Surveys, 66:4 (2011), 767–807
-
A. B. Yakhshimuratov, “Integrirovanie uravneniya Kortevega-de Friza so spetsialnym svobodnym chlenom v klasse periodicheskikh funktsii”, Ufimsk. matem. zhurn., 3:4 (2011), 144–150
-
A. O. Smirnov, G. M. Golovachev, E. G. Amosenok, “Dvukhzonnye 3-ellipticheskie resheniya uravnenii Bussineska i Kortevega–de Friza”, Nelineinaya dinam., 7:2 (2011), 239–256
-
Gaëtan Borot, Bertrand Eynard, “Geometry of Spectral Curves and All Order Dispersive Integrable System”, SIGMA, 8 (2012), 100, 53 pp.
-
Andrei Ya. Maltsev, “Whitham's Method and Dubrovin–Novikov Bracket in Single-Phase and Multiphase Cases”, SIGMA, 8 (2012), 103, 54 pp.
-
Brezhnev Yu.V., “Spectral/Quadrature Duality: Picard-Vessiot Theory and Finite-Gap Potentials”, Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, Contemporary Mathematics, 563, eds. AcostaHumanez P., Finkel F., Kamran N., Olver P., Amer Mathematical Soc, 2012, 1–31
-
A. O. Smirnov, “Solution of a nonlinear Schrödinger equation in the form of two-phase
freak waves”, Theoret. and Math. Phys., 173:1 (2012), 1403–1416
-
Yu. V. Brezhnev, “Elliptic solitons, Fuchsian equations, and algorithms”, St. Petersburg Math. J., 24:4 (2013), 555–574
-
M. M. Matyoqubov, A. B. Yakhshimuratov, “Integration of higher Korteweg-de Vries equation with a self-consistent source in class of periodic functions”, Ufa Math. J., 5:1 (2013), 102–111
-
A. O. Smirnov, “Periodic Two-Phase “Rogue Waves””, Math. Notes, 94:6 (2013), 897–907
-
A. O. Smirnov, G. M. Golovachev, “Trekhfaznye resheniya nelineinogo uravneniya Shredingera v ellipticheskikh funktsiyakh”, Nelineinaya dinam., 9:3 (2013), 389–407
-
A. Badanin, E. Korotyaev, “Third order operator with periodic coefficients on the real axis”, St. Petersburg Math. J., 25:5 (2014), 713–734
-
Kwiatkowski G., Leble S., “Quantum Corrections to I Center Dot (4) Model Solutions and Applications to Heisenberg Chain Dynamics”, Cent. Eur. J. Phys., 11:7 (2013), 887–893
-
B. T. Saparbaeva, “Two-Dimensional Finite-Gap Schrödinger Operators with Elliptic Coefficients”, Math. Notes, 747–749
-
Zhai Yu. Geng X. He G., “Explicit Quasi-Periodic Solutions of the Vakhnenko Equation”, J. Math. Phys., 55:5 (2014), 053512
-
Maltsev A.Ya., “On the Minimal Set of Conservation Laws and the Hamiltonian Structure of the Whitham Equations”, J. Math. Phys., 56:2 (2015), 023510
-
Aleksandr O. Smirnov, Sergei G. Matveenko, Sergei K. Semenov, Elena G. Semenova, “Three-Phase Freak Waves”, SIGMA, 11 (2015), 032, 14 pp.
-
I. Egorova, Z. Gladka, G. Teschl, “On the form of dispersive shock waves of the Korteweg–de Vries equation”, Zhurn. matem. fiz., anal., geom., 12:1 (2016), 3–16
-
A. B. Yakhshimuratov, M. M. Matyokubov, “Integration of loaded Korteweg–de Vries equation in a class of periodic functions”, Russian Math. (Iz. VUZ), 60:2 (2016), 72–76
-
A. E. Mironov, “Self-adjoint commuting differential operators of rank two”, Russian Math. Surveys, 71:4 (2016), 751–779
-
Eynard B., “Counting Surfaces: Crm Aisenstadt Chair Lectures”, Counting Surfaces: Crm Aisenstadt Chair Lectures, Progress in Mathematical Physics, 70, Birkhauser Boston, 2016, 1–414
-
Mironov A.E., Zu D., “Spectral Curve of the Halphen Operator”, Proc. Edinb. Math. Soc., 60:2 (2017), 451–460
-
Rybkin A., “KdV Equation Beyond Standard Assumptions on Initial Data”, Physica D, 365 (2018), 1–11
-
S. M. Grudsky, A. V. Rybkin, “On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg–de Vries Equation”, Math. Notes, 104:3 (2018), 377–394
-
Vladimir P. Kotlyarov, “A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations and their Finite-Gap Solutions”, SIGMA, 14 (2018), 082, 27 pp.
-
Matveev V.B. Smirnov A.O., “Akns and Nls Hierarchies, Mrw Solutions, P-N Breathers, and Beyond”, J. Math. Phys., 59:9, SI (2018), 091419
-
Yu J.-P., Ma W.-X., Sun Y.-L., Khalique Ch.M., “N-Fold Darboux Transformation and Conservation Laws of the Modified Volterra Lattice”, Mod. Phys. Lett. B, 32:33 (2018), 1850409
-
Geng X. Liu W. Xue B., “Finite Genus Solutions to the Coupled Burgers Hierarchy”, Results Math., 74:1 (2019), UNSP 11
-
Wei J. Geng X. Zeng X., “The Riemann Theta Function Solutions For the Hierarchy of Bogoyavlensky Lattices”, Trans. Am. Math. Soc., 371:2 (2019), 1483–1507
-
A. B. Hasanov, M. M. Hasanov, “Integration of the nonlinear Schrödinger equation with an additional term in the class of periodic functions”, Theoret. and Math. Phys., 199:1 (2019), 525–532
-
Albert J.P., “A Uniqueness Result For 2-Soliton Solutions of the Korteweg-de Vries Equation”, Discret. Contin. Dyn. Syst., 39:7 (2019), 3635–3670
-
Marvan M. Pavlov M.V., “Integrable Dispersive Chains and Their Multi-Phase Solutions”, Lett. Math. Phys., 109:5 (2019), 1219–1245
|
Number of views: |
This page: | 1210 | Full text: | 468 | References: | 32 | First page: | 3 |
|