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Additional Reductions in the K -Constrained Modified KP Hierarchy

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Additional reductions are proposed for the modified k-constrained KP hierarchy. As a result, we obtain generalizations of Kaup–Broer system, Korteweg–de-Vries equation, and a modification of the Korteweg–de-Vries equation that belongs to the modified k-constrained KP hierarchy. We also propose a solution generating technique based on binary Darboux transformations for the obtained equations.

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References

  1. Y. Ohta, J. Satsuma, D. Takahashi, and T. Tokihiro, “An elementary introduction to Sato theory,” Prog. Theor. Phys. Suppl., 94, 210–241 (1988).

    Article  MathSciNet  Google Scholar 

  2. L. A. Dickey, “Soliton equations and Hamiltonian systems,” Adv. Ser. Math. Phys., 26 (2003).

  3. M. Blaszak, Multi-Hamiltonian Theory of Dynamical Systems, Springer-Verlag (1998).

  4. J. Sidorenko andW. Strampp, “Symmetry constraints of the KP-hierarchy,” Inverse Problems, 7, L37–L43 (1991).

  5. B.G. Konopelchenko, J. Sidorenko, and W. Strampp, “(1+1)-Dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems,” Phys. Lett. A, 157, 17–21 (1991).

    Article  MathSciNet  Google Scholar 

  6. Y. Cheng and Y. S. Li, “The constraint of the Kadomtsev–Petviashvili equation and its special solutions,” Phys. Lett. A, 157, 22–26 (1991).

    Article  MathSciNet  Google Scholar 

  7. Y. Cheng, “Constrained of the Kadomtsev–Petviashvili hierarchy,” J. Math. Phys., 33, 3774–3787 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  8. Y. Cheng and Y. S. Li, “Constraints of the 2+1 dimensional integrable soliton systems,” J. Phys. A, 25, No. 2, 419–431 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  9. V. K. Melnikov, “On equations for wave interactions,” Lett. Math. Phys., 7, No. 2, 129–136 (1983).

    Article  MathSciNet  Google Scholar 

  10. V. K. Melnikov, “A direct method for deriving a multi-soliton solution for the problem of interaction of waves on the x, y plane,” Comm. Math. Phys., 112, No. 4, 639–652 (1987).

    Article  MathSciNet  Google Scholar 

  11. V. K. Melnikov, “Exact solutions of the Korteweg–de Vries equation with a self-consistent source,” Phys. Lett. A, 128, 488–492 (1988).

    Article  MathSciNet  Google Scholar 

  12. V.G. Samoilenko, “Differential-geometric structure and spectral properties of nonlinear completely integrable dynamical systems of the Mel’nikov type,” Ukr. Math. J., 42, No. 5, 579–583 (1990).

    Article  Google Scholar 

  13. J. Sidorenko and W. Strampp, “Multicomponent integrable reductions in Kadomtsev–Petviashvilli hierarchy,” J. Math. Phys., 34, No. 4, 1429–1446 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Oevel, “Darboux theorems and Wronskian formulas for integrable systems I: Constrained KP flows,” Physica A, 195, 533–576 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  15. Y.-J. Zhang and Y. Cheng, “Solutions for the vector k-constrained KP-hierarchy,” J. Math. Phys., 35, 5869–5884 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  16. W. Oevel and W. Strampp, “Wronskian solutions of the constrained KP hierarchy,” J. Math. Phys., 37, 6213–6219 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  17. H. Aratyn, E. Nissimov, and S. Pacheva, “Constrained KP hierarchy: additional symmetries, Darboux–Backlund solutions and relations to multi-matrix models,” Int. J. Mod. Phys. A, 12, 1265–1340 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Willox, I. Loris, and C. R. Gilson, “Binary Darboux transformations for constrained KP hierarchies,” Inverse Problems, 13, 849–865 (1997).

    Article  MathSciNet  Google Scholar 

  19. A. Kundu, W. Strampp, and W. Oevel, “Gauge transformations of constrained KP flows: new integrable hierarchies,” J. Math. Phys., 36, 2972–2984 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  20. W. Oevel and S. Carillo, “Squared eigenfunction symmetries for soliton equations,” J. Math. Anal. and Appl., 217, 161–199 (1998).

    Article  MathSciNet  Google Scholar 

  21. Yu. Sidorenko, “Transformation operators for integrable hierarchies with additional reductions,” in: Proc. Inst. Math. Nat. Acad. Sci. Ukraine, 43, Pt 1, 352–357 (2002).

  22. Yu. Yu. Berkela and Yu. M. Sydorenko, “Darboux type theorems and transformation operators for nonlocal reduced Hermitian Kadomtsev–Petviashvili hierarchy (Hk-cKP),” Mat. Stud., 25, No. 1, 38–64 (2006).

    MATH  MathSciNet  Google Scholar 

  23. Yu. Yu. Berkela, Integration of Nonlinear Evolution Systems with Nonlocal Constraints, Candidate-Degree Thesis (Physics and Mathematics), Lviv (2005).

  24. Yu. O. Mytropolsky, V. H. Samoilenko, and Yu. M. Sidorenko, “Spatially two-dimensional generalization of the KP hierarchy with nonlocal constraints,” Dop. Nat. Acad. Sci. Ukraine, 9, 19–23 (1999).

    Google Scholar 

  25. A. M. Samoilenko, V. G. Samoilenko, and Yu. M. Sidorenko, “Hierarchy of the Kadomtsev–Petviashvili equations under nonlocal constraints: Multidimensional generalizations and exact solutions of reduced systems,” Ukr. Math. J., 51, No. 1, 86–106 (1999).

    Article  MathSciNet  Google Scholar 

  26. X. J. Liu, Y. B. Zeng, and R. Lin, “A new extended KP hierarchy,” Phys. Lett. A., 372, 3819–3823 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  27. X. J. Liu, R. Lin, B. Jin, and Y.-B. Zeng, “A generalized dressing approach for solving the extended KP and the extended mKP hierarchy,” J. Math. Phys., 50, 053506-1–053506-14 (2009).

  28. Y. Huang, X. Liu, Y. Yao, and Y. Zeng, “A new extended matrix KP hierarchy and its solutions,” Theor. Math. Phys., 167, No. 2, 590–605 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  29. Y.-H. Huang, Y.Q. Yao, and Y. B. Zeng, “A new .γA; σB/-matrix KP hierarchy and its solutions,” Comm. Theor. Phys., 57, No. 4, 515–522 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  30. O. Chvartatskyi and Yu. Sydorenko, “Matrix generalizations of integrable systems with Lax integro-differential representations,” J. Phys.: Conf. Ser., 411, 012010-1– 012010–11 (2013). e-print:http://arxiv.org/abs/1212.3444.

  31. O. Chvartatskyi and Yu. Sydorenko, “A new (1+1)-dimensional matrix KP hierarchy,” Mat. Stud., 39, No. 2, 164–178 (2013). e-print:http://arxiv.org/abs/1303.7064.

  32. O. Chvartatskyi and Yu. Sydorenko, “A new bidirectional generalization of (2+1)-dimensional matrix k-constrained KP hierarchy,” J. Math. Phys., 54, 113508-1–113508-22 (2013). e-print:http://arxiv.org/abs/1303.6510.

  33. V. S. Dryuma, “On the analytic solution of the two-dimensional Korteweg–de Vries equation,” JETP Lett., 19, No. 12, 753–755 (1974).

    Google Scholar 

  34. V. E. Zakharov and A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I,” Funct. Anal. and Appl., 8, No. 3, 226–235 (1974).

    Article  MATH  Google Scholar 

  35. Yu. Sydorenko and O. Chvartatskyi, “Binary transformations of the spatially two-dimensional operators and Lax equations,” Visn. Kyiv Univ. Mech. and Math., 22, 32–35 (2009).

    Google Scholar 

  36. I. Loris and R. Willox, “Symmetry reduction of the BKP hierarchy,” J. Math. Phys., 40, 1420–1431 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  37. J. He, Zh. Wu, and Yi. Cheng, “Gauge transformations for the constrained CKP and BKP hierarchies,” J. Math. Phys., 48, No. 11, 113519-1–13519-16 (2007).

  38. H. -X. Wu, X. -J. Liu, and Y. -B. Zeng, “Two new multi-component BKP hierarchies,” Comm. Theor. Phys., 51, 193–204 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  39. F. A. Khalilov and E. Ya. Khruslov, “Matrix generalization of the modified Korteweg–de Vries equation,” Inverse Problems, 6 (1990).

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Authors and Affiliations

  1. Franko Lviv Nat. Univ., Universytets’ka st., 1, Lviv, 79000, Ukraine

    O. Chvartatskyi & Yu. Sydorenko

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  1. O. Chvartatskyi
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  2. Yu. Sydorenko
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Correspondence to O. Chvartatskyi or Yu. Sydorenko.

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Published in Neliniini Kolyvannya, Vol. 17, No. 3, pp. 419–436, July–September, 2014.

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Chvartatskyi, O., Sydorenko, Y. Additional Reductions in the K -Constrained Modified KP Hierarchy. J Math Sci 212, 349–369 (2016). https://doi.org/10.1007/s10958-015-2669-x

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