Geodesic
A direct algorithm for constructing recursion operators and Lax pairs for integrable models
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 294-312 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator $R$ can be represented as a ratio of the form $R=L_1^{-1}L_2$, where the linear differential operators $L_1$ and $L_2$ are chosen such that the ordinary differential equation $(L_2-\lambda~L_1)U=0$ is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter $\lambda\in\mathbb{C}$. To construct the operator $L_1$, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek $L_2$, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation $L_1\widetilde U=L_2U$ defines a Bäcklund transformation mapping a solution $U$ of the linearized equation to another solution $\widetilde U$ of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.
Mots-clés : Lax pair
Keywords: integrable chain, higher symmetry, invariant manifold, recursion operator. 
@article{TMF_2018_196_2_a6,
     author = {I. T. Habibullin and A. R. Khakimova},
     title = {A~direct algorithm for constructing recursion operators and {Lax} pairs for integrable models},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {294--312},
     year = {2018},
     volume = {196},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a6/}
}
TY  - JOUR
AU  - I. T. Habibullin
AU  - A. R. Khakimova
TI  - A direct algorithm for constructing recursion operators and Lax pairs for integrable models
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2018
SP  - 294
EP  - 312
VL  - 196
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a6/
LA  - ru
ID  - TMF_2018_196_2_a6
ER  - 
%0 Journal Article
%A I. T. Habibullin
%A A. R. Khakimova
%T A direct algorithm for constructing recursion operators and Lax pairs for integrable models
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2018
%P 294-312
%V 196
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a6/
%G ru
%F TMF_2018_196_2_a6
I. T. Habibullin; A. R. Khakimova. A direct algorithm for constructing recursion operators and Lax pairs for integrable models. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 294-312. http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a6/

[1] I. T. Habibullin, A. R. Khakimova, M. N. Poptsova, “On a method for constructing the Lax pairs for nonlinear integrable equations”, J. Phys. A: Math. Theor., 49:3 (2016), 035202, 35 pp. | DOI | MR | Zbl

[2] E. V. Pavlova, I. T. Khabibullin, A. R. Khakimova, “Ob odnoi integriruemoi diskretnoi sisteme”, Differentsialnye uravneniya. Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. matem. i ee pril. Temat. obz., 140, VINITI RAN, M., 2017, 30–42

[3] I. T. Khabibullin, A. R. Khakimova, “Invariantnye mnogoobraziya i pary Laksa dlya integriruemykh nelineinykh tsepochek”, TMF, 191:3 (2017), 369–388 | DOI | MR

[4] I. T. Habibullin, A. R. Khakimova, “On a method for constructing the Lax pairs for integrable models via a quadratic ansatz”, J. Phys. A: Math. Theor., 50:30 (2017), 305206, 19 pp. | DOI | MR | Zbl

[5] N. Kh. Ibragimov, A. B. Shabat, “Evolyutsionnye uravneniya s netrivialnoi gruppoi Li–Beklunda”, Funkts. analiz i ego pril., 14:1 (1980), 25–36 | DOI | MR | Zbl

[6] B. I. Suleimanov, “«Kvantovaya» linearizatsiya uravnenii Penleve kak komponenta ikh $L,A$ par”, Ufimsk. matem. zhurn., 4:2 (2012), 127–135 | MR

[7] V. E. Zakharov, A. B. Shabat, “Skhema integrirovaniya nelineinykh uravnenii matematicheskoi fiziki metodom obratnoi zadachi rasseyaniya. I”, Funkts. analiz i ego pril., 8:3 (1974), 43–53 | DOI | MR | MR | Zbl

[8] V. E. Zakharov, A. B. Shabat, “Integrirovanie nelineinykh uravnenii matematicheskoi fiziki metodom obratnoi zadachi rasseyaniya. II”, Funkts. analiz i ego pril., 13:3 (1979), 13–22 | DOI | MR | Zbl

[9] H. D. Wahlquist, F. B. Estabrook, “Prolongation structures of nonlinear evolution equations”, J. Math. Phys., 16:1 (1975), 1–7 | DOI | MR | Zbl

[10] J. Weiss, M. Tabor, G. Carnevale, “The Painlevé property for partial differential equations”, J. Math. Phys., 24:3 (1983), 522–526 | DOI | MR | Zbl

[11] M. Musette, R. Conte, “Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations”, J. Math. Phys., 32:6 (1991), 1450–1457 | DOI | MR | Zbl

[12] F. W. Nijhoff, A. J. Walker, “The discrete and continuous Painlevé VI hierarchy and the Garnier system”, Glasg. Math. J., 43:A (2001), 109–123 | DOI | MR

[13] A. I. Bobenko, Yu. B. Suris, “Integrable systems on quad-graphs”, Internat. Math. Res. Not., 2002:11 (2002), 573–611 | DOI | MR | Zbl

[14] F. W. Nijhoff, “Lax pair for the Adler (lattice Krichever–Novikov) system”, Phys. Lett. A, 297:1–2 (2002), 49–58, arXiv: nlin/0110027 | DOI | MR | Zbl

[15] R. I. Yamilov, “O klassifikatsii diskretnykh uravnenii”, Integriruemye sistemy, ed. A. B. Shabat, BFAN SSSR, Ufa, 1982, 95–114 | Zbl

[16] P. Xenitidis, “Integrability and symmetries of difference equations: the Adler–Bobenko–Suris case”, Group Analysis of Differential Equations and Integrable Systems (Protaras, Cyprus, October 26–30, 2008), eds. N. Ivanova, C. Sophocleous, R. Popovych, P. Damianou, A. Nikitin, Cyprus Univ., Protaras, Cyprus, 2008, 226–142, arXiv: 0902.3954 | MR

[17] M. Gürses, A. Karasu, V. V. Sokolov, “On construction of recursion operators from Lax representation”, J. Math. Phys., 40:12 (1999), 6473–6490 | DOI | MR | Zbl

[18] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, “Korteweg–de Vries equation and generalizations. VI. Methods for exact solution”, Commun. Pure Appl. Math., 27:1 (1974), 97–133 | DOI | MR | Zbl

[19] I. M. Gelfand, L. A. Dikii, “Drobnye stepeni operatorov i gamiltonovy sistemy”, Funkts. analiz i ego pril., 10:4 (1976), 13–29 | DOI | MR | Zbl

[20] F. Magri, “A simple model of the integrable Hamiltonian equation”, J. Math. Phys., 19:5 (1978), 1156–1170 | DOI | MR

[21] V. V. Sokolov, “O gamiltonovosti uravneniya Krechevera–Novikova”, Dokl. AN SSSR, 277:1 (1984), 48–50 | MR

[22] M. Gürses, A. Karasu, “Integrable KdV systems: recursion operators of degree four”, Phys. Lett. A, 251:4 (1999), 247–249 | DOI | MR | Zbl

[23] S. I. Svinolupov, “Iordanovy algebry i obobschennye uravneniya Kortevega–de Friza”, TMF, 87:3 (1991), 391–403 | DOI | MR | Zbl

[24] J. Krasil'shchik, “Cohomology background in geometry of PDE”, Secondary Calculus and Cohomological Physics, Contemporary Mathematics, 219, eds. M. Henneaux, J. Krasil'shchik, A. Vinogradov, AMS, Providence, RI, 1998, 121–140 | DOI

[25] H. Zhang, G. Z. Tu, W. Oevel, B. Fuchssteiner, “Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure”, J. Math. Phys., 32:7 (1991), 1908–1918 | DOI | MR | Zbl

[26] W. Oevel, H. Zhang, B. Fuchssteiner, “Mastersymmetries and multi-Hamiltonian formulations for some integrable lattice systems”, Progr. Theor. Phys., 81:2 (1989), 294–308 | DOI | MR

[27] P. J. Olver, “Evolution equations possessing infinitely many symmetries”, J. Math. Phys., 18:6 (1977), 1212–1215 | DOI | MR | Zbl

[28] R. I. Yamilov, “Symmetries as integrability criteria for differential difference equations”, J. Phys. A: Math. Gen., 39:45 (2006), R541–R623 | DOI | MR | Zbl

[29] D. Levi, R. Yamilov, “Conditions for the existence of higher symmetries of evolutionary equations on the lattice”, J. Math. Phys., 38:12 (1997), 6648–6674 | DOI | MR | Zbl

[30] F. Khanizade, A. V. Mikhailov, Dzh. P. Vang, “Preobrazovaniya Darbu i rekursionnye operatory dlya differentsialno-raznostnykh uravnenii”, TMF, 177:3 (2013), 387–440 | DOI | DOI | MR | Zbl

[31] R. N. Garifullin, E. V. Gudkova, I. T. Habibullin, “Method for searching higher symmetries for quad-graph equations”, J. Phys. A: Math. Theor., 44:32 (2011), 325202, 16 pp. | DOI | MR

[32] P. E. Hydon, C. M. Viallet, “Asymmetric integrable quad-graph equations”, Appl. Anal., 89:4 (2010), 493–506 | DOI | MR | Zbl

[33] R. Garifullin, I. Habibullin, M. Yangubaeva, “Affine and finite Lie algebras and integrable Toda field equations on discrete space-time”, SIGMA, 8 (2012), 062, 33 pp. | DOI | MR | Zbl

[34] B. A. Dubrovin, V. B. Matveev, S. P. Novikov, “Nelineinye uravneniya tipa Kortevega–de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, UMN, 31:1(187) (1976), 55–136 | DOI | MR | Zbl