Airault, H. Rational solutions of Painleve equations. (English) Zbl 0496.58012 Stud. Appl. Math. 61, 31-53 (1979). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 2 ReviewsCited in 93 Documents MSC: 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 35P25 Scattering theory for PDEs 34A55 Inverse problems involving ordinary differential equations 37A30 Ergodic theorems, spectral theory, Markov operators Keywords:rational solutions of Painleve equations; inverse scattering problems × Cite Format Result Cite Review PDF Full Text: DOI Digital Library of Mathematical Functions: §32.10(ii) Second Painlevé Equation ‣ §32.10 Special Function Solutions ‣ Properties ‣ Chapter 32 Painlevé Transcendents §32.8(i) Introduction ‣ §32.8 Rational Solutions ‣ Properties ‣ Chapter 32 Painlevé Transcendents §32.8(v) Fifth Painlevé Equation ‣ §32.8 Rational Solutions ‣ Properties ‣ Chapter 32 Painlevé Transcendents References: [1] Ablowitz, Nonlinear evolution equations and ordinary differential equations of Painlevé type, Lettere Al Nuovo Cimento. 23 (9) (1978) · doi:10.1007/BF02824479 [2] Ablowitz, Exact linearization of a Painlevé transcendent, Phys. Rev. Lett. 38 pp 20– (1977) · doi:10.1103/PhysRevLett.38.1103 [3] Ince, Ordinary Differential Equations (1944) [4] Gambier, Sur les equations differentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes, Acta Math 33 pp 1– (1910) · JFM 40.0377.02 · doi:10.1007/BF02393211 [5] Airault, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem, Comm. Pure Appl. Math. 30 pp 95– (1977) · Zbl 0338.35024 · doi:10.1002/cpa.3160300106 [6] Ablowitz, Solitons and rational solutions of nonlinear evolution equations, J. Mathematical Phys. 19 (10) pp 2180– (1978) · Zbl 0418.35022 · doi:10.1063/1.523550 [7] Adler, On a class of polynomials connected with the Korteweg-de Vries equation, Communications in Math. Physics 61 (1) (1978) · Zbl 0428.35067 · doi:10.1007/BF01609465 [8] Kruskal, Lectures in Applied Mathematics 15 pp 61– (1972) [9] Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 pp 1202– (1968) · Zbl 0283.35018 · doi:10.1063/1.1664700 [10] Bordag, Two dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett. Phys. Lett. 63A (3) pp 205– (1977) [11] Lamb, Analytical descriptions of ultrashort optical pulse propagation in a resonant medium, Rev. Modern Phys. 43 pp 99– (1971) · doi:10.1103/RevModPhys.43.99 [12] Hirota, A new form of Backlund transformation and its relation to the inverse scattering problem, Progr. Theoret. Phys. 52 pp 1498– (1974) · Zbl 1168.37322 · doi:10.1143/PTP.52.1498 [13] Dodd, Backlund transformations for the sine-Gordon equations, Proc. Roy. Soc. London Ser. A 351 pp 499– (1976) · Zbl 0353.35063 · doi:10.1098/rspa.1976.0154 [14] Bäcklund Transformations · Zbl 0475.58002 [15] Calogero, Bäcklund transformations and functional relations for solutions of nonlinear partial differential equations solvable via the inverse scattering method, Lett. Nuovo Cimento 14 pp 537– (1975) · doi:10.1007/BF02785140 [16] McCoy, Connection between the KdV equation and the two dimensional Ising model, Phys. Lett. 61A (5) pp 283– (1977) · doi:10.1016/0375-9601(77)90613-2 [17] Berezin, Theory of nonstationary finite-amplitude waves in a low-density plasma, Z. Eksper. Teoret. Fiz. 46 pp 1880– (1964) [18] Ablowitz, Asymptotic solutions of the Korteweg-de Vries equation, Studies in Appl. Math. 57 pp 13– (1977) · Zbl 0369.35055 · doi:10.1002/sapm197757113 [19] McKean, The spectrum of Hill’s equation, Invent. Math. 30 pp 217– (1975) · Zbl 0319.34024 · doi:10.1007/BF01425567 [20] Ablowitz, The inverse scattering transform. Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 pp 249– (1974) · Zbl 0408.35068 · doi:10.1002/sapm1974534249 [21] Bateman, Higher transcendental functions 2 (1953) [22] Painlevé, Surles equations differentielles du second order et d’ordre superieur, dont l’integrale générale est uniforme, Acta Math. 25 pp 1– (1902) · JFM 32.0340.01 · doi:10.1007/BF02419020 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.