zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Rational solutions of the Toda lattice equation in Casoratian form. (English) Zbl 1090.37053
Summary: A recursive procedure is presented for constructing rational solutions to the Toda lattice equation through the Casoratian formulation. It allows us to compute a broad class of rational solutions directly, without computing long wave limits in soliton solutions. All rational solutions arising from the Taylor expansions of the generating functions of soliton solutions are special ones of the general class, but only a Taylor expansion containing even or odd powers leads to non-constant rational solutions. A few rational solutions of lower order are worked out.

37K60Lattice dynamics (infinite-dimensional systems)
37K15Integration of completely integrable systems by inverse spectral and scattering methods
Full Text: DOI
[1] Airault, H.; Mckean, H. P.; Moser, J.: Rational and elliptic solutions of the Korteweg--de Vries equation and a related many-body problem. Commun. pure appl. Math. 30, 95-148 (1977) · Zbl 0338.35024
[2] Adler, M.; Moser, J.: On a class of polynomials connected with the Korteweg--de Vries equation. Commun. math. Phys. 61, 1-30 (1978) · Zbl 0428.35067
[3] Strampp, W.; Oevel, W.; Steeb, W. -H.: Similarity, Bäcklund transformations and rational solutions. Lett. math. Phys. 7, 445-452 (1983) · Zbl 0542.35068
[4] Gesztesy, F.; Schweiger, W.: Rational KP and mkp-solutions in Wronskian form. Rep. math. Phys. 30, 205-222 (1991) · Zbl 0766.35045
[5] Ablowitz, M. J.; Satsuma, J.: Solitons and rational solutions of nonlinear evolution equations. J. math. Phys. 19, 2180-2186 (1978) · Zbl 0418.35022
[6] Toda, M.: Vibration of a chain with nonlinear interaction. J. phys. Soc. jpn. 22, 431-436 (1967)
[7] Cârstea, A. S.; Grecu, D.: On a class of rational and mixed soliton-rational solutions of the Toda lattice. Prog. theoret. Phys. 96, 29-36 (1996)
[8] Narita, K.: Solutions for the Mikhailov--Shabat--yamilov difference-differential equations and generalized solutions for the Volterra and the Toda lattice equations. Prog. theoret. Phys. 99, 337-348 (1998)
[9] Nimmo, J. J. C.; Freeman, N. C.: Rational solutions of the Korteweg--de Vries equation in Wronskian form. Phys. lett. A 96, 443-446 (1983) · Zbl 0588.35077
[10] Weiss, J.: Modified equations, rational solutions, and the Painlevé property for the Kadomtsev--Petviashvili and Hirota--satsuma equations. J. math. Phys. 26, 2174-2180 (1985) · Zbl 0588.35020
[11] Hu, X. B.; Clarkson, P. A.: Rational solutions of a differential-difference KdV equation, the Toda equation and the discrete KdV equation. J. phys. A: math. Gen. 28, 5009-5016 (1995) · Zbl 0868.35132
[12] Medina, E.; Marı\acute{}n, M. J.: Rational solutions of the KP equation through symmetry reductions: coherent structures and other solutions. Inverse problems 17, 985-998 (2001) · Zbl 0988.35149
[13] Nimmo, J. J. C.: Soliton solution of three differential-difference equations in Wronskian form. Phys. lett. A 99, 281-286 (1983) · Zbl 1168.34365
[14] Tamizhmani, T.; Vel, S. Kanaga; Tamizhmani, K. M.: Wronskian and rational solutions of the differential-difference KP equation. J. phys. A: math. Gen. 31, 7627-7633 (1998) · Zbl 0931.35154
[15] Ma, W. X.: Complexiton solutions to the Korteweg--de Vries equation. Phys. lett. A 301, 35-44 (2002) · Zbl 0997.35066
[16] Ma WX, Maruno K. Complexiton solutions of the Toda lattice equation. 2003, preprint
[17] Wu, H.; Zhang, D. J.: Mixed rational-soliton solutions of two differential-difference equations in Casorati determinant form. J. phys. A: math. Gen. 36, 4867-4873 (2003) · Zbl 1033.37038
[18] Toda, M.: Nonlinear waves and solitons. (1989) · Zbl 0705.35120
[19] Hirota, R.; Satsuma, J.: A variety of nonlinear network equations generated from the Bäcklund transformation for the Toda lattice. Prog. theoret. Phys. suppl., No. 59, 64-100 (1976) · Zbl 1079.35536
[20] Matveev, V. B.; Salle, M. A.: Darboux transformations and solitons. (1991) · Zbl 0744.35045
[21] Ma WX, You Y. Solving the Korteweg--de Vries equation by its bilinear form: Wronskian solutions. 2003, preprint
[22] Maruno K, Ma WX, Oikawa M. Generalized Casorati determinant and positon-negaton type solutions of the Toda lattice equation. 2003, preprint · Zbl 1053.37063
[23] Ablowitz, M. J.; Cornille, H.: On solutions of the Korteweg--de Vries equation. Phys. lett. A 72, 277-280 (1979)
[24] Stahlhofen, A. A.; Matveev, V. B.: Positons for the Toda lattice and related spectral problems. J. phys. A: math. Gen. 28, 1957-1965 (1995) · Zbl 0843.58071
[25] Kajiwara, K.; Noumi, M.; Yamada, Y.: Q-Painlevé systems arising from q-KP hierarchy. Lett. math. Phys. 62, 259-268 (2002) · Zbl 1030.37045
[26] Clarkson, P. A.; Mansfield, E. L.: The second Painlevé equation, its hierarchy and associated special polynomials. Nonlinearity 16, R1-R26 (2003)