Factorization of matrix differential
operators and Darboux-like transformations

Yu. M. Sydorenko

A factorization of a matrix differential operator of general form into elementary factors is constructed in explicit form. A notion of simplest elementary Darboux transformations is introduced. The theory of general Darboux type transformations for matrix evolutionary differential operators of arbitrary order is developed. Binary Darboux type transformations for such operators are also constructively represented.

1. Baecklund A.V. Zur Theorie der partielen Differential gleichungen erster Ordnung // Math. Ann. – V.17. – 1880. – P. 285–328.

2. Darboux G. Le cons sur la théorie générale de surface et les applications geometriques du calcul infinitesimal II. Gauthiers-Villars, Paris, 1889.

3. Crum M.M. Associated Sturm-Liouville systems // Quart. J. Math. Oxford. – 1955. – V.2, № 6. – P.121–127.

4. Matveev V.B. Darboux transformations and explicit solutions of the Kadomtsev–Petviashvily equation depending on the functional parameters // Letter in Mathematical Physics. – 1979. – V.3. – P.213–216.

5. Matveev V.B., Salle M.A. Darboux transformations and solitons. – Springer-Verlag, Berlin Heidelberg. – 1991. – 120 p.

6. Sidorenko Yu.M. Nonlocal reductions in system which are integrable by inverse scattering method, In: Nonlinear boundary-value problems of mathematical physics and their applications, Kyiv: Institute of Mathematics, 1998, P.199–202.

7. Samoilenko V.H., Sidorenko Yu.M., Berkela Yu.Yu. A new class of C-integrable systems. Direct linearization of nonlinear models, In: Nonlinear boundary-value problems of mathematical physics and their applications, Kyiv: Institute of Mathematics, 1998, P.189–192.

8. Sidorenko Yu.M., Hvozdova Ye.V. Binary trasformation of general Lax-Zakharov-Shabat equations // In: Nonlinear boundary-value problems of mathematical physics and their applications, Kyiv: Institute of Mathematics, 1999. – P.220–224.

9. Sidorenko Yu. Binary transformation and (2+1)-dimensional integrable systems // Ukr. math. journ. – 2002. – V.52, № 11. – P.1531–1550

10. Konopelchenko B., Sidorenko Yu., Strampp W. (1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems // Phys. Lett. A. – 1991. – V. 157. – P.17–21.

11. Oevel W., Sidorenko Yu., Strampp W. Hamiltonian structures of the Melnickov system and its reductions // Inverse Problems. – 1993. – V. 9. – P. 737–747.

12. Sidorenko Yu., Strampp W. Multicomponents integrable reductions in Kadomtsev-Petviashvilli hierarchy // J. Math. Phys. – 1993. –V.34, № 4. – P. 1429–1446.

13. Sidorenko Yu. KP-hierachy and (1+1)-dimensional multicomponent integrable systems // Ukr. math. journ. – 1993. – V.25, № 1. – P. 91–104.

14. Cheng Yi. Constrained of the Kadomtsev-Petviashvili hierarchy // J. Math. Phys. – 1992. – V. 33. – P. 3774.

15. Cheng Yi, Zhang Y.J. Solutions for the vector $k$-constrainted KP-hierarchy // J. Math. Phys. -- 1994. -- V.35, № 11.

16. Samoilenko A.M., Samoilenko V.H., Sidorenko Yu.M. Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced systems // Ukr. Math. Jour. – 1999. – V.51, № 1. – P.78–97.

17. Sidorenko Yu.M. Method of integration of the Lax equations with nonlocal reductions // Proceedings of NAS of Ukraine. – 1999. – № 8. – P. 19–23.

18. Berkela Yu.Yu., Sidorenko Yu.M. The exact solutions of some multicomponent integrable models // Mat. Studii. – 2002. – V.17, № 1. – P.47–58.

19. Oevel W. Darboux theorems and Wronskian formulas for integrable systems I: Constrained KP flows // Physica A. – 1993. – P.533–576.

20. Oevel W., Strampp W. Wronskian solutions of the constrained Kadomtsev-Petviashvili hierarhy // J.Math.Phys. – 1996. – V.37, № 12. – P.6213–6219.

21. Willox R., Loris I., Gilson C.R. Binary Darboux transformations for constrained KP hierarchies // Inverse problems. – 1997. – V.13. – P.849–865.

22. Chau L.L., Shaw J.C., Tu M.H. Solving the constrained KP hierarchy by gauge transformations// J.Math.Phys. – 1997. – V.38, № 8. – P.4128–4137.

23. Sidorenko Yu. Transformation operators for integrable hierarchies with additional reductions // Proceedings of Institute of Math. of NAS of Ukraine. – 2002.– V.43, Part 1. – P.352–357.

24. Sydorenko Yu.M., Berkela Yu.Yu. Integration of nonlinear space-twodimensional Heisenberg equations // Mat. Studii. – 2002. – V.18, № 1. – P.57–68.

25. Matveev V.B., Salle M.A. New families of the explicit solutions of the Kadomtsev-Petviashvili equation and their application to Johnson equatio, In: “Some topic on Inverse Problems", ed. P.C.Sabatier (Singapoore: World Scientific).– 1998.

26. Nimmo J.J.C. Darboux transformations for a two-dimensional Zakharov-Shabat/ AKNS spectral problem // Inverse Problems.– 1992.– V. 8. – P.219–243.

27. Guil F., Manas M., Alvarez G. The Hopf-Cole transformation and KP equation // Phys. Lett. A. – 1994. – V.190. – P.49–52.

28. Guil F., Manas M. Darboux transformation for the Davey-Stewartson equations // Phys. Lett. A. – 1996. – V.217. – P.1–6.

29. Boiti M., Pempinelli F., Pogrebkov A.K., Polivanov M.C. New features of Baecklund and Darboux transformations in 2+1 dimensions // Inverse Problems. – 1991. – V.7.– P.43–56.

30. Sakhnovich A. Iterated Baecklund–Darboux transformation and transfer matrix-function (nonisosrectral case) // Chaos, Solitons & Fractals. – 1996. – V.7, № 8. – P.1251–1259.

31. Sakhnovich A. Canonical systems and transfer matrix-functions // Proc. Amer. Math. Soc. – 1997. – V.125, № 5. – P.1451–1455.

32. Sakhnovich A., Zubelli J.P. Bundle bispectrality for matrix differential equations // Integr. equ. oper. theory. – 2001. – V.41. – P.472–496.

33. Sakhnovich A. Generalized Baecklund-Darboux transformation: spectral properties and nonlinear equations // Journ. of Math. Anal. and Applic. – 2001. – V.262. – P.274–306

34. Nimmo J.J.C., Gilson K.R., Ohta Y. Application of Darboux transformation to the self-dual Yang-Mills equations // Theor. and Math. Phys. – 2000. – V.122, № 2. – P.284–293. (in Russian)

35. Tsarev S.P. Factorizations of linear partial differential operators and Darboux method of integration of nonlinear equations with partial derivatives // Theor. and Math. Phys. – 2000. – V.122, № 1. – P.144–160. (in Russian)

36. Weiss J. Periodic fixed points of Baeklund transformations and the KdV equation // Journ. Math. Phys. – 1986. – V.27. – P.2647–2656.

37. Weiss J. Periodic fixed points of Baeklund transformations // Journ. Math. Phys. – 1987. – V.28. – P.2025–2039.

38. Veselov A.P., Shabat A.B. Dressing chain and spectral theory of the Shroedinger operator // Func. Anal. and Appl. – 1993.– V.27. – № 2.– P.1–21 (in Russian).

39. Takasaki K. Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains // arXiv:nlin.SI/0206049. – V.2, 30 Jun 2002. – 56 p.

	Factorization of matrix differential operators and Darboux-like transformations
Author	Yu. M. Sydorenko matmod@franko.lviv.ua Faculty of Mechanics and Mathematics, Lviv Ivan Franko National University
Abstract	A factorization of a matrix differential operator of general form into elementary factors is constructed in explicit form. A notion of simplest elementary Darboux transformations is introduced. The theory of general Darboux type transformations for matrix evolutionary differential operators of arbitrary order is developed. Binary Darboux type transformations for such operators are also constructively represented.
Keywords	factorization of a matrix differential operator, elementary factors, explicit form, Darboux transformations, matrix evolutionary differential operators
DOI	doi:10.30970/ms.19.2.181-192
Reference	1. Baecklund A.V. Zur Theorie der partielen Differential gleichungen erster Ordnung // Math. Ann. – V.17. – 1880. – P. 285–328. 2. Darboux G. Le cons sur la théorie générale de surface et les applications geometriques du calcul infinitesimal II. Gauthiers-Villars, Paris, 1889. 3. Crum M.M. Associated Sturm-Liouville systems // Quart. J. Math. Oxford. – 1955. – V.2, № 6. – P.121–127. 4. Matveev V.B. Darboux transformations and explicit solutions of the Kadomtsev–Petviashvily equation depending on the functional parameters // Letter in Mathematical Physics. – 1979. – V.3. – P.213–216. 5. Matveev V.B., Salle M.A. Darboux transformations and solitons. – Springer-Verlag, Berlin Heidelberg. – 1991. – 120 p. 6. Sidorenko Yu.M. Nonlocal reductions in system which are integrable by inverse scattering method, In: Nonlinear boundary-value problems of mathematical physics and their applications, Kyiv: Institute of Mathematics, 1998, P.199–202. 7. Samoilenko V.H., Sidorenko Yu.M., Berkela Yu.Yu. A new class of C-integrable systems. Direct linearization of nonlinear models, In: Nonlinear boundary-value problems of mathematical physics and their applications, Kyiv: Institute of Mathematics, 1998, P.189–192. 8. Sidorenko Yu.M., Hvozdova Ye.V. Binary trasformation of general Lax-Zakharov-Shabat equations // In: Nonlinear boundary-value problems of mathematical physics and their applications, Kyiv: Institute of Mathematics, 1999. – P.220–224. 9. Sidorenko Yu. Binary transformation and (2+1)-dimensional integrable systems // Ukr. math. journ. – 2002. – V.52, № 11. – P.1531–1550 10. Konopelchenko B., Sidorenko Yu., Strampp W. (1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems // Phys. Lett. A. – 1991. – V. 157. – P.17–21. 11. Oevel W., Sidorenko Yu., Strampp W. Hamiltonian structures of the Melnickov system and its reductions // Inverse Problems. – 1993. – V. 9. – P. 737–747. 12. Sidorenko Yu., Strampp W. Multicomponents integrable reductions in Kadomtsev-Petviashvilli hierarchy // J. Math. Phys. – 1993. –V.34, № 4. – P. 1429–1446. 13. Sidorenko Yu. KP-hierachy and (1+1)-dimensional multicomponent integrable systems // Ukr. math. journ. – 1993. – V.25, № 1. – P. 91–104. 14. Cheng Yi. Constrained of the Kadomtsev-Petviashvili hierarchy // J. Math. Phys. – 1992. – V. 33. – P. 3774. 15. Cheng Yi, Zhang Y.J. Solutions for the vector $k$-constrainted KP-hierarchy // J. Math. Phys. -- 1994. -- V.35, № 11. 16. Samoilenko A.M., Samoilenko V.H., Sidorenko Yu.M. Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced systems // Ukr. Math. Jour. – 1999. – V.51, № 1. – P.78–97. 17. Sidorenko Yu.M. Method of integration of the Lax equations with nonlocal reductions // Proceedings of NAS of Ukraine. – 1999. – № 8. – P. 19–23. 18. Berkela Yu.Yu., Sidorenko Yu.M. The exact solutions of some multicomponent integrable models // Mat. Studii. – 2002. – V.17, № 1. – P.47–58. 19. Oevel W. Darboux theorems and Wronskian formulas for integrable systems I: Constrained KP flows // Physica A. – 1993. – P.533–576. 20. Oevel W., Strampp W. Wronskian solutions of the constrained Kadomtsev-Petviashvili hierarhy // J.Math.Phys. – 1996. – V.37, № 12. – P.6213–6219. 21. Willox R., Loris I., Gilson C.R. Binary Darboux transformations for constrained KP hierarchies // Inverse problems. – 1997. – V.13. – P.849–865. 22. Chau L.L., Shaw J.C., Tu M.H. Solving the constrained KP hierarchy by gauge transformations// J.Math.Phys. – 1997. – V.38, № 8. – P.4128–4137. 23. Sidorenko Yu. Transformation operators for integrable hierarchies with additional reductions // Proceedings of Institute of Math. of NAS of Ukraine. – 2002.– V.43, Part 1. – P.352–357. 24. Sydorenko Yu.M., Berkela Yu.Yu. Integration of nonlinear space-twodimensional Heisenberg equations // Mat. Studii. – 2002. – V.18, № 1. – P.57–68. 25. Matveev V.B., Salle M.A. New families of the explicit solutions of the Kadomtsev-Petviashvili equation and their application to Johnson equatio, In: “Some topic on Inverse Problems", ed. P.C.Sabatier (Singapoore: World Scientific).– 1998. 26. Nimmo J.J.C. Darboux transformations for a two-dimensional Zakharov-Shabat/ AKNS spectral problem // Inverse Problems.– 1992.– V. 8. – P.219–243. 27. Guil F., Manas M., Alvarez G. The Hopf-Cole transformation and KP equation // Phys. Lett. A. – 1994. – V.190. – P.49–52. 28. Guil F., Manas M. Darboux transformation for the Davey-Stewartson equations // Phys. Lett. A. – 1996. – V.217. – P.1–6. 29. Boiti M., Pempinelli F., Pogrebkov A.K., Polivanov M.C. New features of Baecklund and Darboux transformations in 2+1 dimensions // Inverse Problems. – 1991. – V.7.– P.43–56. 30. Sakhnovich A. Iterated Baecklund–Darboux transformation and transfer matrix-function (nonisosrectral case) // Chaos, Solitons & Fractals. – 1996. – V.7, № 8. – P.1251–1259. 31. Sakhnovich A. Canonical systems and transfer matrix-functions // Proc. Amer. Math. Soc. – 1997. – V.125, № 5. – P.1451–1455. 32. Sakhnovich A., Zubelli J.P. Bundle bispectrality for matrix differential equations // Integr. equ. oper. theory. – 2001. – V.41. – P.472–496. 33. Sakhnovich A. Generalized Baecklund-Darboux transformation: spectral properties and nonlinear equations // Journ. of Math. Anal. and Applic. – 2001. – V.262. – P.274–306 34. Nimmo J.J.C., Gilson K.R., Ohta Y. Application of Darboux transformation to the self-dual Yang-Mills equations // Theor. and Math. Phys. – 2000. – V.122, № 2. – P.284–293. (in Russian) 35. Tsarev S.P. Factorizations of linear partial differential operators and Darboux method of integration of nonlinear equations with partial derivatives // Theor. and Math. Phys. – 2000. – V.122, № 1. – P.144–160. (in Russian) 36. Weiss J. Periodic fixed points of Baeklund transformations and the KdV equation // Journ. Math. Phys. – 1986. – V.27. – P.2647–2656. 37. Weiss J. Periodic fixed points of Baeklund transformations // Journ. Math. Phys. – 1987. – V.28. – P.2025–2039. 38. Veselov A.P., Shabat A.B. Dressing chain and spectral theory of the Shroedinger operator // Func. Anal. and Appl. – 1993.– V.27. – № 2.– P.1–21 (in Russian). 39. Takasaki K. Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains // arXiv:nlin.SI/0206049. – V.2, 30 Jun 2002. – 56 p.
Pages	181-192
Volume	19
Issue	2
Year	2003
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Factorization of matrix differential operators and Darboux-like transformations