References
M. Adler and J. Moser, “On a class of polynomials connected with the Korteweg–de Vries equation,” Comm. Math. Phys., 61, 1–30 (1978).
Z. S. Agranovich and V. A. Marchenko, The Inverse Problem of Scattering Theory, Gordon and Breach Science Publishers, New York–London (1960).
H. Airault, H.P. McKean, and J. Moser, “Rational and elliptic solutions of the Korteweg–de Vries equation and a related many-body problem,” Comm. Pure Appl. Math., 30, 95–148 (1977).
C. Athorne and J. J. C. Nimmo, “On the Moutard transformation for integrable partial differential equations,” Inverse Problems, 7, 809–826 (1991).
M. M. Crum, “Associated Sturm–Liouville systems,” Quart. J. Math. Oxford Ser. (2), 6, 121–127 (1955).
G. Darboux, “Sur une proposition relative aux équations linéarires,” C. R. Acad. Sci. Paris, 94, 1456–1459 (1882).
P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford (1935).
B. A. Dubrovin, I.M. Krichever, and S.P. Novikov, “The Schrödinger equation in a periodic field and Riemann surfaces,” Sov. Math., Dokl., 17, 947–952 (1976).
L. D. Faddeev, “The inverse problem in the quantum theory of scattering,” J. Math. Phys., 4, 72–104 (1963).
L. D. Faddeev, “Properties of the S-matrix of the one-dimensional Schrödinger equation,” Tr. Mat. Inst. Steklova, 73, 314–336 (1964).
L. D. Faddeev, “The inverse problem in the quantum theory of scattering. II,” Current Problems in Mathematics, 3, 93–180 (1974).
I. M. Gel’fand and B. M. Levitan, “On the determination of a differential equation from its spectral function,” Amer. Math. Soc. Transl. (2), 1, 253–304 (1955).
P. G. Grinevich and S. V. Manakov, “Inverse scattering problem for the two-dimensional Schrödinger operator, the \( \bar{\partial } \)-method and nonlinear equations,” Funct. Anal. Appl., 20, 94–103 (1986).
P. G. Grinevich and S.P. Novikov, “Two-dimensional “inverse scattering problem” for negative energies and generalized-analytic functions. I: Energies below the ground state,” Funct. Anal. Appl., 22, No. 1, 19–27 (1988).
H. C. Hu, S.Y. Lou, and Q.P. Liu, “Darboux transformation and variable separation approach: the Nizhnik–Novikov–Veselov equation,” Chinese Phys. Lett., 20, 1413–1415 (2003).
L. Infeld and T.E. Hull, “The factorization method,” Rev. Modern Phys., 23, 21–68 (1951).
C. E. Kenig, G. Ponce, and L. Vega, “Well–posedness of the inital value problem for the Korteweg–de Vries equation,” J. Amer. Math. Soc., 4, 323–347 (1991).
S. V. Manakov, “The method of the inverse scattering problem, and two-dimensional evolution equations,” Uspehi Mat. Nauk, 31, No. 5 (191), 245–246 (1976).
V. A. Marchenko, Sturm–Liouville Operators and Their Applications, Naukova Dumka, Kiev (1977).
V. B. Matveev and M.A. Salle, Darboux Transformations and Solitons, Springer, Berlin (1991).
A. Menikoff, “The existence of unbounded solutions of the Korteweg–de Vries equation,” Comm. Pure Appl. Math., 25, 407–432 (1972).
T. Moutard, “Sur la construction des équations de la forme \( \frac{1}{z}\frac{{{d^2}z}}{{dxdy}} = \lambda \left( {x,y} \right) \), qui admettent une intégrale générale explicite,” J. École Polytechnique, 45, 1–11 (1878).
R. G. Novikov and G.M. Khenkin, “The \( \bar{\partial } \)-equation in the multidimensional inverse scattering problem,” Russian Math. Surveys, 42, No. 3, 109–180 (1987).
S.P. Novikov and A.P. Veselov, “Exactly solvable two-dimensional Schrödinger operators and Laplace transformations,” Trans. Amer. Math. Soc., 179, 109–132 (1997).
E. Schrödinger, “Further studies on solving eigenvalue problems by factorization,” Proc. Roy. Irish Acad. Sect. A., 46, 183-206 (1940).
E. Schrödinger, “Method of determining quantum-mechanical eigenvalues and eigenfunctions,” Proc. Roy. Irish Acad. Sect. A., 46, 9–16 (1940).
E. Schrödinger, “The factorization of the hypergeometric equation,” Proc. Roy. Irish Acad. Sect. A., 47, 53-54 (1941).
I. A. Taĭmanov and S.P. Tsarëv, “Two-dimensional Schrödinger operators with rapidly decaying rational potential and multidimensional L 2-kernel,” Russian Math. Surveys, 62, No. 3, 631–633 (2007).
I. A. Taĭmanov and S.P. Tsarëv, “Two-dimensional Schrödinger rational solitons and their blowup via the Moutard transformation,” Theor. Math. Phys., 157, No. 2, 1525–1541 (2008).
I. A. Taĭmanov and S.P. Tsarëv, “Blowing up solutions of the Novikov–Veselov equation,” Dokl. Math., Math. Phys., 77, No. 3, 467–468 (2008).
A.P. Veselov and S.P. Novikov, “Finite-zone, two-dimensional Schrödinger operators. Explicit formulas and evolution equations,” Sov. Math., Dokl., 30, 588–591 (1984).
A.P. Veselov and S.P. Novikov, “Finite-zone, two-dimensional Schrödinger operators. Potential operators,” Sov. Math., Dokl., 30, 705–708 (1984).
H. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the Korteweg–de Vries equation,” Phys. Rev. Lett., 31, 1386–1390 (1973).
V. F. Zaharov, S. V. Manakov, S.P. Novikov, and L.P. Pitaevskiĭ, Theory of Solitons. The Method of the Inverse Problem [in Russian], Nauka, Moscow (1980).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 35, Proceedings of the Fifth International Conference on Differential and Functional Differential Equations. Part 1, 2010.
Rights and permissions
About this article
Cite this article
Taimanov, I.A., Tsarëv, S.P. On the Moutard transformation and its applications to spectral theory and Soliton equations. J Math Sci 170, 371–387 (2010). https://doi.org/10.1007/s10958-010-0092-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-010-0092-x