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Auto-Bäcklund transformations and superposition formulas for solutions of Drinfeld-Sokolov systems. (English) Zbl 1311.35252
Summary: The paper is devoted to constructing Auto-Bäcklund transformations (ABT) and superposition formulas for the solutions of the Drinfeld-Sokolov (DS) systems. The transformations are derived from pairs of differential substitutions relating different systems of the DS type. The nonlinear superposition formulas for solutions of the DS systems are obtained from the assumption of commutativity of the Bianchi diagram. We indicate a seed solution for each system which can be used to generate multi-soliton solutions. As an application of the superposition formulas we construct two-soliton solutions for each of the DS systems.

MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35A30 Geometric theory, characteristics, transformations in context of PDEs
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References:
 [1] V.G. Drinfeld, V.V. Sokolov, Equations of Korteweg-de Vries type, and simple Lie algebras, Dokl. Akad. Nauk SSSR 258 (1) (1981) 11-16 (in Russian). [2] V.G. Drinfeld, V.V. Sokolov, Proceedings of S.L. Sobolev, Seminar, Novosibirsk 2 (5) (1981) (in Russian). [3] Ito, M., An extension of nonlinear evolution equations of the K-dv (mk-dv) type to higher orders, J. phys. soc. jpn., 49, 2, 771-778, (1980) · Zbl 1334.35282 [4] Hirota, R.; Satsuma, J., Soliton solutions of a coupled Korteweg-de Vries equation, Phys. lett. A, 85, 8-9, 407-408, (1981) [5] Wilson, G., The affine Lie algebra $$C_2^{(1)}$$ and an equation of Hirota and Satsuma, Phys. lett. A, 89, 7, 332-334, (1982) [6] Karasu, A.; Sakovich, S.Yu., Bäcklund transformation and special solutions for Drinfeld-Sokolov-Satsuma-Hirota system of coupled equations, J. phys. A: math. gen., 34, 36, 7355-7358, (2001) · Zbl 0983.35116 [7] Verhoeven, C.; Musette, M., Grammian N-soliton solutions of a coupled KdV system, J. phys. A: math. gen., 34, 49, 721-725, (2001) · Zbl 0987.35145 [8] A.G. Meshkov, M.Ju. Balakhnev, Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions, SIGMA 4, Paper 018 (2008), arXiv:0802.1253. · Zbl 1142.37045 [9] A.G. Meshkov, I.V. Kulemin, To the classification of integrable systems in 1+1 dimensions. Symmetry in nonlinear mathematical physics, vol. 1, 2 (Kyiv, 1997), 115-121, Natl. Acad. Sci. Ukraine, Inst. Math., Kiev, 1997. · Zbl 0947.35160 [10] V.E. Zakharov, The inverse scattering method, in: Solitons (Ed.), R.K. Bullough, J.P. Caudrey (Springer, 1980); Top. Curr. Phys. 17, pp. 243-285. [11] A.G. Meshkov, V.V. Sokolov, Integrable evolution equations on the N-dimensional sphere, Comm. Math. Phys. 232 (1) (2002) 1-18. · Zbl 1012.37049 [12] A.P. Fordy, A historical introduction to solitons and Bäcklund transformations. Harmonic maps and integrable systems, 7-28, Aspects Math., E23, Vieweg, Braunschweig, 1994. · Zbl 0810.35101 [13] Wahlquist, H.D.; Estabrook, F.B., Bäcklund transformation for solutions of the Korteweg-de Vries equation, Phys. rev. lett., 31, 23, 1386-1390, (1973) [14] Kodama, Y.; Wadati, M., Theory of canonical transformations for nonlinear evolution equations, I. progr. theoret. phys., 56, 6, 1740-1755, (1976) · Zbl 1079.35504 [15] Adler, V.E.; Marikhin, V.G.; Shabat, A.B., Lagrangian chains and canonical Bäcklund transformations, Theoret. math. phys., 129, 2, 1448-1465, (2001) · Zbl 1029.37042 [16] Balakhnev, M.Ju., Superposition formulas for vector generalizations of the mkdv equation, Math. notes, 82, 4, 448-450, (2007) · Zbl 1156.35456 [17] Mikhailov, A.V.; Shabat, A.B.; Sokolov, V.V., The symmetry approach to classification of integrable equations, (), 115-184 · Zbl 0741.35070 [18] M.Ju. Balakhnev, I.V. Kulemin, Differential substitutions for the third order evolution systems, Differential Equations and Control Processes, Electronic J., 1 (2002), http:www.neva.ru/journal (in Russian). [19] Demskoi, D.K., On application of Liouville type equations to constructing Bäcklund transformations, J. nonlinear math. phys., 14, 1, 147-156, (2007) · Zbl 1165.35465
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