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Solving AKNS equations via Jacobi inversion problem

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Abstract

By using Lax representation, we study the separation of variables forx– andt n– finitedimensional integrable Hamiltonian system (FDIHS) obtained from the factorization of AKNS hierarchy. Then the separability ofx– andt n–FDIHS and the factorization of AKNS hierarchy give rise to the Jacobi inversion problem for soliton equations in AKNS hierarchy. By a standard Jacobi inversion technique, the soliton equations can be solved in terms of Riemann theta function.

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References

  1. S.P. Novikov. A Method for Solving the Periodic Problem for the KdV Equation and Its Generalization. In: Topic in Current Physics 17, R. Bullough and P. Caudrey, eds., Spinger, New York, 1980. 325–338

    Google Scholar 

  2. A.C. Newell, M. Tabor, Zeng Yunbo. A Unified Approach to Painleve’ Expansion.Physica D, 1987, 29: 1–68

    Article  MATH  Google Scholar 

  3. E.K. Sklyanin. Separation of Variables.Prog. Theor. Phys. Suppl., 1995, 118: 35–60

    Article  Google Scholar 

  4. E.G. Kalnins, V.B. Kuznetsov, Willard Miller, Jr. Quadrics on Complex Riemannian Spaces of Constant Curvature, Separation of Variables, and the Gaudin Magnet.J. Math. Phys., 1994, 35, 1710–1731

    Article  MATH  Google Scholar 

  5. J.C. Eilbeck, V.Z. Enol’skii, V.B. Kuznetsov, D.V. Leykin. Linearr-matrix Algebra for Systems Separable in Parabolic Coordinates.Phys. Lett. A 1993, 180: 208–214

    Article  Google Scholar 

  6. J. Harnad, P. Winternitz. Classical and Quantum Integrable Systems in\(\widetilde{gl}\)(2) and separation of variables.Commun. Math. Phys., 1995, 172: 263–285

    Article  MATH  Google Scholar 

  7. Cao Cewen. Nonlinearization of the Lax System for AKNS Hierarchy.Science in China A, 1990, 33: 528–536

    MATH  Google Scholar 

  8. Zeng Yunbo. An Approach to the Deduction of Finite-dimensional Integrability from Infinite-dimensional Integrability.Phys. Lett. A, 1991, 160: 541–547

    Article  Google Scholar 

  9. Zeng Yunbo. New Factorization of the Kaup-Newell Hierarchy.Physica D, 1994, 73: 171–188

    Article  MATH  Google Scholar 

  10. Zeng Yunbo, Li Yishen. The Deduction of the Lax Representation for Constrained Flows from the Adjoint Representation.J. Phys. A, 1993, 26: L273-L278

    Article  MATH  Google Scholar 

  11. P.P. Kulish, S. Rauch-Wojciechowski, A.V. Tsiganov. Restricted Flows of the KdV Hierarchy andr-matrix Formalism.Modern Phys. Lett. A, 1994, 9: 2063–2073

    Article  MATH  Google Scholar 

  12. Zeng Yunbo. The Separability and Dynamicalr-matrix for the Constrained Flows of the Jaulent Miodek Hierarchy.Phys. Lett. A, 1996, 216, 26–32

    Article  MATH  Google Scholar 

  13. Zeng Yunbo. A Farmily of Separable Integrable Hamiltonian Systems and Their Dynamicalr-matrix.Inverse Problems, 1996, 12: 797–809

    Article  Google Scholar 

  14. M.J. Ablowitz, H. Segur. Solitons and Inverse Scattering Transformation. SIAM, Philadelpia, 1981

    Google Scholar 

  15. Zeng Yunbo, Li Yishen. The Lax Representation and Darboux Transformation for Constrained Flows of AKNS Hierarchy.Acta Mathematica Sinica (New Series), 1996, 12(2): 217–224

    Article  MATH  Google Scholar 

  16. V.I. Arnold. Mathematical Method of Classical Mechanics. Springer, New York, 1978

    Google Scholar 

  17. B.A. Dubrovin. Theta Function and Nonlinear Equations.Russ. Math. Surv., 1981, 36: 11–92

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, China

    Zeng Yunbo

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  1. Zeng Yunbo
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This works is supported by National Basic Research Project “Nonlinear Science”

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Yunbo, Z. Solving AKNS equations via Jacobi inversion problem. Acta Mathematicae Applicatae Sinica 15, 337–344 (1999). https://doi.org/10.1007/BF02684033

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  • DOI: https://doi.org/10.1007/BF02684033

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