×

zbMATH — the first resource for mathematics

The Laplace method, algebraic curves, and nonlinear equations. (English. Russian original) Zbl 0583.35086
Funct. Anal. Appl. 18, 210-223 (1984); translation from Funkts. Anal. Prilozh. 18, No. 3, 43-56 (1984).
An algebro-geometric generalization of the Laplace method [E. Goursat, ”Cours d’analyse mathématique”. Tome II (1949; Zbl 0034.341)] is developed. It allows to find integrable equations of the form \(y''=(u(x)+ax+b)y\) where a,b are constants and u(x) a periodic function tending to a finite-gap potential as \(| x| \to \infty\). The construction is based on the concept of a Laplace type differential. A multiparameter generalization of the latter results in new classes of exact solutions to the Kadomtsev-Petviashvili equation.
Reviewer: S.Duzhin

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35C05 Solutions to PDEs in closed form
34A30 Linear ordinary differential equations and systems, general
35R10 Functional partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. P. Novikov, ”Two-dimensional Schrödinger operators in periodic fields,” Itogi Nauki i Tekhniki, Sov. Probl. Mat.,23, 3-32 (1983).
[2] F. Calogero and A. Degasperis, ”Inverse spectral problem for the one-dimensional Schrödinger equation with an additional linear potential,” Lett. Nuovo Cimento,23, 143-149 (1978). · doi:10.1007/BF02763080
[3] F. Calogero and A. Degasperis, ”Solution by the spectral transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation,” Lett. Nuovo Cimento,23, 150-154 (1978). · doi:10.1007/BF02763081
[4] F. Calogero and A. Degasperis, Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations, Vol. I, North-Holland, Amsterdam (1982). · Zbl 0501.35072
[5] Li Yishen, ”One special inverse problem of the second order differential equation on the whole real axis,” Chinese Ann. Math.,2, 147-156 (1981). · Zbl 0485.34009
[6] S. Graffi and E. Harrell, ”Inverse scattering for the one-dimensional Stark effect and application to the cylindrical KdV equation,” Ann. Inst. H. Poincaré,36, No. 1, 41-58 (1982). · Zbl 0506.35079
[7] E. Goursat, Cours d’Analyse Mathematique, Tome II, Gauthier-Villars, Paris (1933).
[8] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, ”Nonlinear equations of Korteweg?de Vries type, finite-zone linear operators, and Abelian varieties,” Usp. Mat. Nauk,31, No. 1, 55-136 (1976). · Zbl 0326.35011
[9] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, The Theory of Solitons. The Method of the Inverse Problem [in Russian], Nauka, Moscow (1980). · Zbl 0598.35002
[10] I. M. Krichever, ”Methods of algebraic geometry in the theory of nonlinear equations,” Usp. Mat. Nauk,32, No. 6, 183-208 (1977). · Zbl 0372.35002
[11] I. M. Krichever and S. P. Novikov, ”Holomorphic bundles over algebraic curves, and nonlinear equations,” Usp. Mat. Nauk,35, No. 6, 47-68 (1980). · Zbl 0501.35071
[12] B. A. Dubrovin, ”Theta-functions and nonlinear equations,” Usp. Mat. Nauk,36, No. 2, 11-80 (1981). · Zbl 0478.58038
[13] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, ”Topological and algebraic geometry methods in contemporary mathematical physics. II,” Soviet Scientific Reviews. Math. Phys. Reviews, Vol. 3, OPA, Amsterdam (1982). · Zbl 0534.58002
[14] I. M. Krichever, ”Nonlinear equations and elliptic curves,” Itogi Nauki i Tekhniki, Sov. Probl. Mat.,23, 79-136 (1983). · Zbl 0595.35087
[15] B. A. Dubrovin, ”Matricial finite-zone operators,” Itogi Nauki i Tekhniki, Sov. Probl. Mat.,23, 33-78 (1983).
[16] I. M. Krichever and S. P. Novikov, ”Holomorphic bundles over Riemann surfaces and the Kadomtsev?Petviashvili equation. I,” Funkts. Anal. Prilozhen.,12, No. 4, 41-52 (1978). · Zbl 0393.35061
[17] I. M. Krichever and S. P. Novikov, ”Holomorphic bundles and nonlinear equations. Finitezone solutions of rank 2,” Dokl. Akad. Nauk SSSR,247, No. 1, 33-37 (1979).
[18] V. E. Zakharov and A. B. Shabat, ”A plan for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I,” Funkts. Anal. Prilozhen.,8, No. 3, 43-53 (1974). · Zbl 0303.35024
[19] I. M. Krichever, ”An algebraic-geometric construction of the Zakharov-Shabat equations and their periodic solutions,” Dokl. Akad. Nauk SSSR,227, No. 2, 291-294 (1976). · Zbl 0361.35007
[20] I. M. Krichever, ”Integration of nonlinear equations by the methods of algebraic geometry,” Funkts. Anal. Prilozhen.,11, No. 1, 15-31 (1977). · Zbl 0346.35028
[21] G. Springer, Introduction to Riemann Surfaces, Addison-Wesley, Reading (1957). · Zbl 0078.06602
[22] I. V. Cherednik, ”On conditions for reality in finite-zone integration,” Dokl. Akad. Nauk SSSR,252, No. 5, 1104-1108 (1980). · Zbl 0491.35044
[23] I. M. Krichever, ”An analogue of the d’Alembert formula for the equations of a principal chiral field and the sine-Gordon equation,” Dokl. Akad. Nauk SSSR,253, No. 2, 288-292 (1980). · Zbl 0496.35073
[24] A. N. Leznov and M. N. Saveliev, ”On the two-dimensional system of differential equations,” Commun. Math. Phys.,74, 111-119 (1980). · Zbl 0429.35063 · doi:10.1007/BF01197753
[25] P. Mansfield, ”Solutions of Toda lattice,” Preprint Cambridge Univ., CB 39 EW (1982). · Zbl 0488.73054
[26] A. V. Mikhailov, ”On the integrability of the two-dimensional generalization of the Toda chain,” Pis’ma Zh. Eksp. Teor. Fiz.,30, 443-448 (1978).
[27] E. Date and S. Tanaka, ”Exact solutions for the periodic Toda lattice,” Progr. Theor. Phys.,53, 267-273 (1976). · Zbl 1109.37307
[28] I. M. Krichever, ”Algebraic curves and nonlinear difference equations,” Usp. Mat. Nauk,33, No. 4, 215-216 (1978). · Zbl 0382.39003
[29] A. R. Its and V. B. Matveev, ”On a class of solutions of the Korteweg?de Vries equation,” in: Problems of Mathematical Physics [in Russian], Vol. 8, Leningrad State Univ. (1978).
[30] J. D. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Math.,352, Springer-Verlag, Berlin (1973). · Zbl 0281.30013
[31] H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vols. I and II, McGraw-Hill, New York (1953).
[32] L. D. Landau and E. M. Lifshits (Lifshitz), Quantum Mechanics. Non-Relativistic Theory, Addison-Wesley, Reading (1958).
[33] V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973).
[34] S. V. Manakov, ”On complete integrability and stochastization in discrete dynamical systems,” Zh. Eksp. Teor. Fiz.,67, No. 2, 543-555 (1974).
[35] I. M. Krichever, ”The algebraic-geometric spectral theory of the Schrödinger difference operator and the Peierls model,” Dokl. Akad. Nauk SSSR,265, No. 5, 1054-1058 (1982). · Zbl 0527.39001
[36] L. A. Bordag and V. B. Matveev, ”Darboux transformation and explicit solutions of the cylindrical KdV,” Preprint LPTH E No. 6, Leipzig (1979).
[37] V. S. Dryuma, ”Analytic solutions of the axisymmetric Korteweg?de Vries equation,” Izv. Akad. Nauk Moldav. SSR,3, 297-301 (1976).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.