A new (1+1)-dimensional matrix k-constrained KP hierarchy

Author O. I. Chvartatskyi, Yu. M. Sydorenko
y_sydorenko@franko.lviv.ua, alex.chvartatskyy@gmail.com
Ivan Franko National University of Lviv

Abstract We introduce a new generalization of matrix (1+1)-dimensional $k$-constrained KP hierarchy. The new hierarchy contains matrix generalizations of stationary DS systems, (2+1)-dimensional modified Korteweg-de Vries equation and the Nizhnik equation. A binary Darboux transformation method is proposed for integration of systems from this hierarchy.
Keywords integrable systems; constraints of KP hierarchy; stationary Davey-Stewartson system
Reference 1. L.A. Dickey, Soliton equations and hamiltonian systems. – Advanced Series in Mathematical Physics, 2nd ed. World Scientific, River Edge, NJ, 2003, V.26.

2. V.E. Zakharov, A.B. Shabat, Integration of nonlinear equations of mathematical physics by the method of inverse scattering, Funct. Anal. Appl., 8 (1974), ¹3, 226–235.

3. S.P. Novikov, S.V. Manakov, L.P. Pitaevskij, V.E. Zakharov, Theory of solitons. The inverse scattering methods. – Transl. from the Russian, Contemporary Soviet Mathematics. New York–London: Plenum Publishing Corporation. Consultants Bureau, 1984.

4. R.K. Bullough, P.J. Caudrey, Solitons. – Springer-Verlag, Berlin, 1980.

5. V.A. Marchenko, Nonlinear equations and operator algebras. – Dordrecht, Boston, Lancaster, Tokyo, Reidel, 1988.

6. V.B. Matveev, Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation, depending on functional parameters, Lett. in Math. Phys., 3 (1979), 213–216.

7. V.B. Matveev, M.A. Salle, Darboux transformations and solitons. – Berlin Heidelberg, Springer-Verlag, 1991.

8. E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Operator approach to the Kadomtsev-Petviashvili equation. Transformation groups for soliton equations. III, J. Phys. Soc. Jpn., 50 (1981), 3806–3812.

9. E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equations – Euclidean Lie algebras and reduction of the KP hierarchy, Publ. Res. Inst. Math. Sci., 18 (1982), 1077–1110.

10. M. Sato, Y. Sato, Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, North-Holland Math. Stud., 81 (1983), 259–271.

11. M. Jimbo, T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. Res. Inst. Math. Sci., 19 (1983), 943–1002.

12. Y. Ohta, J. Satsuma, D. Takahashi, T. Tokihiro, An elementary introduction to Sato theory, Prog. Theor. Phys. Suppl., 94 (1988), 210–250.

13. V.K. Melnikov, On equations for wave interactions, Lett. Math. Phys., 7 (1983), ¹2, 129–138.

14. V.K. Melnikov, On equations integrable by the inverse scattering method, Preprint JINR P2-85-958, Dubna, 1985. (in Russian)

15. V.K. Melnikov, A direct method for deriving a multi-soliton solution for the problem of interaction of waves on the x,y plane, Commun. Math. Phys., 112 (1987), 639–652.

16. V.K. Melnikov, Integration method of the Korteweg-de Vries equation with a self-consistent with a selfconsistent source, Phys. Lett. A, 128 (1988), 493–496.

17. J. Sidorenko, W. Strampp, Symmetry constraints of the KP hierarchy, Inverse Problems, 7 (1991), L37– L43.

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

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

20. Y. Cheng, Constraints of the Kadomtsev–Petviashvili hierarchy, J. Math. Phys., 33 (1992), 3774–3782.

21. Y. Cheng, Y.S. Li, Constraints of the 2+1 dimensional integrable soliton systems, J. Phys. A., 25 (1992), 419–431.

22. J. Sidorenko, W. Strampp, Multicomponent integrable reductions in Kadomtsev-Petviashvilli hierarchy, J. Math. Phys., 34 (1993), ¹4, 1429–1446.

23. W. Oevel, Darboux Theorems and Wronskian formulas for integrable systems I: Constrained KP Flows, Physica A, 195 (1993), 533–576.

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

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

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

27. L.-L. Chau, J.-C. Shaw, M.-H. Tu, Solving the constrained KP Hierarchy by Gauge transformations, J. Math. Phys., 38 (1997), ¹8, 4128–4137.

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

29. A. Kundu, W. Strampp, W. Oevel, Gauge transformations of constrained KP flows: New integrable Hierarchies, J. Math. Phys., 36 (1995), 2972–2984.

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

31. Yu.O. Mytropolsky, V.H. Samoilenko, Yu.M. Sidorenko, Spatially two-dimensional generalization of Kadomtsev–Petviashvili hierarchy with nonlocal constraints, Proceedings of NSA of Ukraine, 8 (1999), 19–23.

32. A.M. Samoilenko, V.G. Samoilenko, Yu.M. Sidorenko, Hierarchy of the Kadomtsev–Petviashvili equations under nonlocal constraints: Many–dimensional generalizations and exact solutions of reduced systems, Ukr. Math. Journ., 51 (1999), ¹1, 86–106.

33. Yu.Yu. Berkela, Integration of nonlinear evolution systems with nonlocal constraints, Ph.D. thesis, Ivan Franko National University of Lviv, 2005. (in Ukrainian)

34. Yu.Yu. Berkela, Yu.M. Sidorenko, The exact solutions of some multicomponent integrable models, Mat. Stud., 17 (2002), ¹1, 47–58.

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

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

37. B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15 (1970), 539–541.

38. V.S. Dryuma, On the analytic solution of the two-dimensional Korteweg-de Vries equation, JETP Lett., 19 (1974), ¹12, 753–755.

39. Yu.M. Sydorenko, O.I. Chvartatskyi, Matrix generalizations of integrable systems with Lax integrodifferential representations, J. Phys.: Conf. Ser., 411 (2013), 012010, 1–11, http:// arxiv.org/abs/ 1212.3444.

40. A. Dimakis, F. Muller-Hoissen, Multicomponent Burgers and KP hierarchies, and solutions from a matrix linear system, Symmetry, Integrability and Geometry: Methods and Applications, 5 (2009), 002, 1–18.

41. C.R. Gilson, S.R. Macfarlane, Dromion solutions of noncommutative Davey-Stewartson equations, J. Phys. A: Math. Theor, 42 (2009), ¹23, 235202, 1–20.

42. C.R. Gilson, J.J.C. Nimmo, C.M. Sooman, Matrix Solutions of a Noncommutative KP Equation and a noncommutative mKP equation, Theor. Math. Phys., 159 (2009), ¹3, 796–805.

43. C.R. Gilson, M. Hamanaka, J.J.C. Nimmo, Backlund transformations for non-commutative anti-self-dual Yang-Mills equations, Glasgow Mathematical Journal A, 51 (2009), 83–93.

44. L.P. Nizhnik, Integration of multidimensional nonlinear equations by the inverse problem method, Dokl. Akad. Nauk SSSR, 254 (1980), 332–335. (in Russian)

45. Yu. Sydorenko, O. Chvartatskyi, Binary transformations for spatially two-dimensional operators and Lax equations, Visn. Kyiv Univ.: Mechanics and Mathematics, 22 (2009), 32–35. (in Ukranian)

46. Yu. Sydorenko, O. Chvartatskyi, Integration of scalar Kadomtsev-Petviashvili kierarchy by the method of integral Darboux-like transformations, Visn. Lviv Univ. Ser: mech.-mat., 75 (2011), 181–225. (in Ukrainian)

Pages 164-177
Volume 39
Issue 2
Year 2013
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML