|
SIGMA 2 (2006), 063, 10 pages nlin.SI/0606071
https://doi.org/10.3842/SIGMA.2006.063
The Painlevé Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients
Tadashi Kobayashi a and Kouichi Toda b
a) High-Functional Design G, LSI IP Development
Div., ROHM CO., LTD., 21, Saiin Mizosaki-cho, Ukyo-ku, Kyoto 615-8585, Japan
b) Department of Mathematical Physics, Toyama Prefectural University,
Kurokawa 5180, Imizu, Toyama, 939-0398, Japan
Received November 30, 2005, in final form June 17, 2006; Published online June 30, 2006
Abstract
The general KdV equation (gKdV) derived by T. Chou is
one of the famous (1 + 1) dimensional soliton equations with
variable coefficients. It is well-known that the gKdV equation
is integrable. In this paper a higher-dimensional gKdV equation,
which is integrable in the sense of the Painlevé test, is
presented. A transformation that links this equation to the
canonical form of the Calogero-Bogoyavlenskii-Schiff equation
is found. Furthermore, the form and similar transformation for the
higher-dimensional modified gKdV equation are also obtained.
Key words:
KdV equation with variable-coefficients; Painlevé test; higher-dimensional integrable systems.
pdf (867 kb)
ps (632 kb)
tex (1070 kb)
References
- Lamb Jr.G.L., Elements of soliton theory, New York, Wiley, 1980.
- Akhmediev N.N., Ankiewicz A., Solitons: nonlinear pulses and
beams, London, Chapman & Hall, 1997.
- Infeld E., Rowlands G., Nonlinear waves, solitons and chaos, 2nd
ed., Cambridge, Cambridge University Press, 2000.
- Zakharov V.E. (Editor),
What is integrability?, Berlin, Springer-Verlag, 1991.
- Ablowitz M.J., Segur H., Solitons and the inverse scattering
transform, Philadelphia, Society for Industrial & Applied
Mathematics, 1981.
- Jeffrey A., Kawahara T., Asymptotic methods of nonlinear wave
theory, London, Pitman Advanced Publ., 1982.
- Drazin P.G., Johnson R.S., Solitons: an introduction, Cambridge,
Cambridge University Press, 1989.
- Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution
equations and inverse scattering, Cambridge, Cambridge University
Press, 1991.
- Hirota R., The direct methods in soliton theory, Cambridge,
Cambridge University Press, 2004.
- Lax P.D., Integrals of nonlinear equations of evolution and
solitary waves, Comm. Pure Appl. Math., 1968, V.21,
467-490.
- Calogero F., Degasperis A., Spectral transform and solitons I,
Amsterdam, Elsevier Science, 1982.
- Blaszak M., Multi-Hamiltonian theory of dynamical dystems,
Berlin, Springer-Verlag, 1998.
- Dickey L.A., Soliton equations and Hamiltonian systems, Singapore,
World Scientific, 1990.
- Rogers C., Schief W.K., Bäcklund and Darboux transformations,
Cambridge, Cambridge University Press, 2002.
- Ramani A., Dorizzi B., Grammaticos B., Painlevé conjecture
revisited, Phys. Rev. Lett., 1982, V.49, 1539-1541.
- Weiss J., Tabor M., Carnevale G., The Painlevé property for
partial differential equations, J. Math. Phys., 1983,
V.24, 522-526.
- Gibbon J.D., Radmore P., Tabor M., Wood D., Painlevé property
and Hirota's method, Stud. Appl. Math., 1985, V.72, 39-64.
- Steeb W.H., Euler N., Nonlinear evolution equations and Painlevé
test, Singapore, World Scientific, 1989.
- Ramani A., Gramaticos B., Bountis T., The Painlevé property and
singularity analysis of integrable and non-integrable systems,
Phys. Rep., 1989, V.180, 159-245.
- Chowdhury A.R., Painlevé analysis and its applications, New
York, Chapman & Hall, 1999.
- Conte R. (Editor), The Painlevé property one century later, New
York, Springer-Verlag, 2000.
- Korteweg D.J., de Vries F., On the change of form of long waves
advancing in a rectangular canal, and on a new type of long
stationary waves, Philos. Mag., 1895, V.39, 422-443.
- Kadomtsev B.B., Petviashvili V.I., On the stability of solitary
waves in weakly dispersing media, Dokl. Akad. Nauk SSSR,
1970, V.15, 539-541.
- Calogero F., Degasperis A., Exact solution via the spectral
transform of a generalization with linearly x-dependent
coefficients of the modified Korteweg-de Vries equation,
Lett. Nuovo Cimento, 1978, V.22, 270-279.
- Brugarino T., Pantano P., The integration of Burgers and
Korteweg-de Vries equations with nonuniformities, Phys.
Lett. A, 1980, V.80, 223-224.
- Oevel W., Steeb W.-H., Painlevé analysis for a time-dependent
Kadomtsev-Petviashvili equation, Phys. Lett. A, 1984,
V.103, 239-242.
- Steeb W.-H., Louw J.A., Parametrically driven sine-Gordon equation
and Painlevé test, Phys. Lett. A, 1985, V.113, 61-6.
- Chou T., Symmetries and a hierarchy of the general KdV equation,
J. Phys. A: Math. Gen., 1987, V.20, 359-366.
- Chou T., Symmetries and a hierarchy of the general modified KdV
equation, J. Phys. A: Math. Gen., 1987, V.20, 367-374.
- Joshi N., Painlevé property of general variable-coefficient
versions of the Korteweg-de Vries and non-linear Schrödinger
equations, Phys. Lett. A, 1987, V.125, 456-460.
- Hlavatý L., Painlevé analysis of nonautonomous evolution
equations, Phys. Lett. A, 1988, V.128, 335-338.
- Brugarino T., Painlevé property, auto-Bäcklund transformation,
Lax pairs, and reduction to the standard form for the Korteweg-de
Vries equation with nonuniformities, J. Math. Phys., 1989,
V.30, 1013-1015.
- Chan W.L., Li K.S., Non-propagating solitons of the
non-isospectral and variable coefficient modified KdV
equation, J. Phys., 1989, V.27, 883-902.
- Brugarino T., Greco A.M., Painlevé analysis and reducibility to
the canonical form for the generalized Kadomtsev-Petviashvili
equation, J. Math. Phys., 1991, V.32, 69-72.
- Klimek M., Conservation laws for a class of nonlinear equations
with variable coefficients on discrete and noncommutative
spaces, J. Math. Phys., 2002, V.43, 3610-3635,
math-ph/0011030.
- Gao Y.-T., Tian B., On a variable-coefficient modified KP
equation and a generalized variable-coefficient KP equation
with computerized symbolic computation, Internat. J. Modern
Phys. C, 2001, V.12, 819-833.
- Gao Y.-T., Tian B., Variable-coefficient balancing-act
algorithm extended to a variable-coefficient MKP model for the
rotating fluids, Internat. J. Modern Phys. C, 2001, V.12,
1383-1389.
- Wang M., Wang Y., Zhou Y., An auto-Bäcklund transformation and
exact solutions to a generalized KdV equation with variable
coefficients and their applications, Phys. Lett. A,
2002, V.303, 45-61.
- Cascaval R.C.,
Variable coefficient Korteveg-de Vries equations and wave
propagation in elastic tubes,
in Evolution Equations, Lecture Notes in Pure and Appl. Math., Vol. 234, New York, Dekker, 2003,
57-69.
- Blanchet A., Dolbeault J., Monneau R., On the one-dimensional
parabolic obstacle problem with variable coefficients, in
Elliptic and Parabolic Problems, Progr. Nonlinear
Differential Equations Appl., Vol. 63, Basel, Birkhauser, 2005,
59-66, math.AP/0410330.
- Fokas A.S., Boundary value problems for linear PDEs with variable
coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng.
Sci., 2004, V.460, N 2044, 1131-1151, math.AP/0412029.
- Kobayashi T., Toda K., Extensions of nonautonomous nonlinear
integrable systems to higher dimensions, in Proceedings of 2004
International Symposium on Nonlinear Theory and its Applications,
2004, V.1, 279-282.
- Robbiano L., Zuily C., Strichartz estimates for Schrödinger
equations with variable coefficients, Mem. Soc. Math. Fr.
(N.S.), 2005, N 101-102, math.AP/0501319.
- Loutsenko I., The variable coefficient Hele-Shaw problem,
integrability and quadrature identities, math-ph/0510070.
- Senthilvelan M., Torrisi M., Valenti A., Equivalence
transformations and differential invariants of a generalized
nonlinear Schrödinger equation, nlin.SI/0510065.
- Miura R.M., Gardner C.S., Kruskal M.D., Korteweg-de Vries
equation and generalizations. II. Existence of conservation laws
and constants of motion, J. Math. Phys., 1968, V.9,
1204-1209.
- Kobayashi T., Toda K., Nonlinear integrable equations with
variable coefficients from Painlevé test and their exact
solutions, in Proceedings of the 10th International Conference
"Modern Group Analysis" (2004, Larnaca, Cyprus), 2005, 214-221.
- Kobayashi T., Toda K., A generalized KdV-family with variable
coefficients in (2 + 1) dimensions, IEICE Transactions
on Fundamentals of Electronics, Communications and Computer
Sciences, 2005, V.E88-A, 2548-2553.
- Calogero F., A method to generate solvable nonlinear evolution
equations, Lett. Nuovo Cim., 1975, V.14, 443-447.
- Bogoyavlenskii O.I., Overturning solitons in new two-dimensional
integrable equations, Math. USSR-Izv., 1990, V.34,
245-259.
- Schiff J., Integrability of Chern-Simons-Higgs vortex
equations and a reduction of the selfdual Yang-Mills equations to
three-dimensions, NATO ASI Ser. B, Vol. 278, New York,
Plenum, 1992.
- Yu S.-J., Toda K., Sasa N., Fukuyama T., N soliton solutions to
the Bogoyavlenskii-Schiff equation and a quest for the soliton
solution in (3+1) dimensions, J. Phys. A: Math. Gen.,
1998, V.31, 3337-3347.
- Toda K., Yu S.-J., Fukuyama T., The Bogoyavlenskii-Schiff
hierarchy and integrable equations in (2 + 1) dimensions,
Rep. Math. Phys., 1999, V.44, 247-254.
- Ablowitz M.J., Kaup D.J., Newell A.C., Segur H., The inverse
scattering transform-Fourier analysis for nonlinear problems,
Stud. Appl. Math., 1974, V.53, 249-315.
- Clarkson P.A., Mansfield E.L., On a shallow water wave equation,
Nonlinearity, 1994, V.7, 975-1000, solv-int/9401003.
- Clarkson P.A., Gordoa P.R., Pickering A., Multicomponent equations
associated to non-isospectral scattering problems,
Inverse Problems, 1997, V.13, 1463-1476.
- Yu S.-J., Toda K., Lax pairs, Painlevé properties and exact
solutions of the Calogero-Korteweg-de Vries equation and a new
(2 + 1)-dimensional equation, J. Nonlinear Math. Phys.,
2000, V.7, 1-13, math.AP/0001188.
- Toda K., Extensions of soliton equations to non-commutative (2 +1) dimensions, in JHEP Proceedings of Workshop on Integrable
Theories, Solitons and Duality, PR-HEP, 2003, unesp2002/038,
10 pages.
- Gordoa P.R., Pickering A., Nonisospectral scattering problems: a
key to integrable hierarchies, J. Math. Phys., 1999, V.40,
5749-5786.
- Estévez P.G., A nonisospectral problem in (2+1) dimensions
derived from KP, Inverse Problems, 2001, V.17, 1043-1052.
- Bogoyavlenskii O.I., Breaking solitons. III, Math. USSR-Izv., 1991, V.36, 129-137.
- Gilson C., Pickering A., Factorization and Painlevé analysis of
a class of nonlinear third-order partial differential equations,
J. Phys. A: Math. Gen., 1995, V.28, 2871-2888.
- Hone A.N.W., Painlevé tests, singularity structure and
integrability, Report UKC/IMS/03/33, IMS, University of Kent, UK,
2003, nlin.SI/0502017.
- Estévez P.G., Prada J., Hodograph transformations for a
Camassa-Holm hierarchy in (2 + 1) dimensions, J. Phys. A:
Math. Gen., 2005, V.38, 1287-1297, nlin.SI/0412019.
- Gordoa P.R., Pickering A., Senthilvelan M., A note on the
Painlevé analysis of a (2 + 1) dimensional Camassa-Holm
equation, Chaos Solitons Fractals, 2006, V.28, 1281-1284,
nlin.SI/0511025.
- Strachan I.A.B., A new family ofi ntegrable models in (2+1)
dimensions associated with Hermitian symmetric spaces, J.
Math. Phys., 1992, V.33, 2477-2482.
- Strachan I.A.B., Some integrable hierarchies in (2+1) dimensions
and their twistor description, J. Math. Phys., 1993, V.34,
243-259.
- Jiang Z., Bullough R.K., Integrability and a new breed of solitons
of an NLS type equation in (2 + 1) dimensions, Phys. Lett.
A, 1994, V.190, 249-254.
- Kakei S., Ikeda T., Takasaki K., Hierarchy of (2 +1)-dimensional nonlinear Schrödinger equation, self-dual
Yang-Mills equation, and toroidal Lie algebras, Ann. Henri
Poincaré, 2002, V.190, 817-845, nlin.SI/0107065.
- Kobayashi T., Toda K., Extensions of nonautonomous nonlinear
integrable systems to higher dimensions, in preparation.
|
|