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Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I

Please always quote using this URN: urn:nbn:de:0297-zib-8391
  • We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov--Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of $2^{\mbox{\scriptsize nd }}$order systems with a $3^{\mbox{\scriptsize rd }}$order or a $4^{\mbox{\scriptsize th }}$order symmetry and $3^{\mbox{\scriptsize rd }}$order systems with a $5^{\mbox{\scriptsize th }}$order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.

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Metadaten
Author:Takayuki Tsuchida, Thomas Wolf
Document Type:ZIB-Report
PACS-Classification:00.00.00 GENERAL / 02.00.00 Mathematical methods in physics / 02.30.-f Function theory, analysis / 02.30.Ik Integrable systems
00.00.00 GENERAL / 02.00.00 Mathematical methods in physics / 02.30.-f Function theory, analysis / 02.30.Jr Partial differential equations
00.00.00 GENERAL / 05.00.00 Statistical physics, thermodynamics, and nonlinear dynamical systems (see also 02.50.-r Probability theory, stochastic processes, and statistics) / 05.45.-a Nonlinear dynamics and chaos (see also section 45 Classical mechanics of discrete systems; for chaos in fluid dynamics, see 47.52.+j) / 05.45.Yv Solitons (see 52.35.Sb for solitons in plasma; for solitons in acoustics, see 43.25.Rq-in Acoustics Appendix; see 42.50.Md, 42.65.Tg, 42.81.Dp for solitons in optics; see also 03.75.Lm in matter waves; for solitons in space plasma physics, see 94.05.Fg; f
Date of first Publication:2005/01/13
Series (Serial Number):ZIB-Report (05-05)
ArXiv Id:http://arxiv.org/abs/nlin.SI/0412003
ZIB-Reportnumber:05-05
Published in:Appeared in: J. Phys. A: Math. Gen. 38 (2005) 7691-7733
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