Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T12:35:05.040Z Has data issue: false hasContentIssue false
  • English
  • Français

Binary constrained flows and separation of variables for soliton equations

Published online by Cambridge University Press:  17 February 2009

Wen-Xiu Ma  and
Yunbo Zeng
Wen-Xiu Ma
Affiliation:
Department of Mathematics, University of South Florida, Tampa, FL 33620-5700, USA; e-mail: [email protected].
Yunbo Zeng
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In contrast to mono-constrained flows with N degrees of freedom, binary constrained flows of soliton equations, admitting 2 × 2 Lax matrices, have 2N degrees of freedom. Currently existing methods only enable Lax matrices to yield the first N pairs of canonical separated variables. An approach for constructing the second N pairs of canonical separated variables with N additional separated equations is introduced. The Jacobi inversion problems for binary constrained flows are then established. Finally, the separability of binary constrained flows together with the factorization of soliton equations by the spatial and temporal binary constrained flows leads to the Jacobi inversion problems for soliton equations.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 129 - 139
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Ablowitz, M. J., Kaup, D. J., Newell, A. C. and Segur, H., “The inverse scattering transform—Fourier analysis for nonlinear problems”, Studies in Appl. Math. 53 (1974) 249–315.CrossRefGoogle Scholar
[2]Antonowicz, M. and Rauch-Wojciechowski, S., “Soliton hierarchies with sources and Lax representation for restricted flows”, Inverse Problems 9 (1993) 201–215.CrossRefGoogle Scholar
[3]Dubrovin, B. A., “Theta-functions and nonlinear equations”, Russian Math. Survey 36 (1981) 11–92.CrossRefGoogle Scholar
[4]Eilbeck, J. C., Enol'skii, V. Z., Kuznetsov, V. B. and Tsiganov, A. V., “Linear r-matrix algebra for classical separable systems”, J. Phys. A: Math. Gen. 27 (1994) 567–578.CrossRefGoogle Scholar
[5]Harnad, J. and Wintemitz, P., “Classical and quantum integrable systems in and separation of variables”, Commun. Math. Phys. 172 (1995) 263–285.CrossRefGoogle Scholar
[6]Kulish, P. P., Rauch-Wojciechowski, S. and Tsiganov, A. V., “Stationary problems for equation of the KdV type and dynamical r-matrices”, J. Math. Phys. 37 (1996) 3463–3482.CrossRefGoogle Scholar
[7]Kuznetsov, V. B., “Quadrics on real Riemannian spaces of constant curvature: separation of variables and connection with Gaudin magnet”, J. Math. Phys. 33 (1992) 3240–3254.CrossRefGoogle Scholar
[8]Li, Y. and Ma, W. X., “Binary linearization of AKNS spectral problem under higher-order symmetry constraints”, Chaos, Solitons and Fractals 11 (2000) 697–710.Google Scholar
[9]Ma, W. X., “New finite-dimensional integrable systems by symmetry constraint of the KdV equations”, J. Phys. Soc. Jpn. 64 (1995) 1085–1091.CrossRefGoogle Scholar
[10]Ma, W. X., “Symmetry constraint of MKdV equations by binary nonlinearization”, PhysicaA 219 (1995) 467–481.CrossRefGoogle Scholar
[11]Ma, W. X. and Fuchssteiner, B., “Binary linearization of the Lax pairs”, in Nonlinear Physics (eds. Alfinito, E., Boiti, M., Martina, L. and Pempinelli, F.), (World Scientific, Singapore, 1996) 217–224.Google Scholar
[12]Ma, W. X., Fuchssteiner, B. and Oevel, W., “A 3 × 3 matrix spectral problem for AKNS hierarchy and its binary nonlinearization”, PhysicaA 233 (1996) 331–354.CrossRefGoogle Scholar
[13]Ma, W. X. and Strampp, W., “An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems”, Phys. Lett.A 185 (1994) 277–286.CrossRefGoogle Scholar
[14]Sklyanin, E. K., “Separation of variables—new trends. Quantum field theory, integrable models and beyond”, Prog. Theor Phys. Suppl. 118 (1995) 35–60.CrossRefGoogle Scholar
[15]Zeng, Y., “Separation of variables for the constrained flows”, J. Math. Phys. 38 (1997) 321–329.CrossRefGoogle Scholar
[16]Zeng, Y., “Using factorization to solve soliton equations”, J. Phys. A: Math. Gen. 30 (1997) 3719–3724.CrossRefGoogle Scholar
[17]Zeng, Y. and Li, Y., “The deduction of the Lax representation for constrained flows from the adjoint representation”, J. Phys. A: Math. Gen. 26 (1993) L273–L278.CrossRefGoogle Scholar
[18]Zeng, Y. and Ma, W. X., “Separation of variables for soliton equations via their binary constrained flows”, J. Math. Phys. 40 (1999) 6526–6557.CrossRefGoogle Scholar