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Integrable theory of the perturbation equations. (English) Zbl 1080.37578
Summary: An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations, such as hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures, and provides us with a method of generating hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones. The resulting perturbation equations give rise to a sort of integrable coupling of soliton equations. Two examples (MKdV hierarchy and KP equation) are carefully carried out.

##### MSC:
 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 35Q53 KdV equations (Korteweg-de Vries equations) 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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##### References:
 [1] Karpman, V.I., Soliton evolution in the presence of perturbation, Phys. scr., 20, 462-478, (1979) · Zbl 1063.35531 [2] Keener, J.P.; McLaughlin, D.W., Solitons under perturbations, Phys. rev. A, 16, 777-790, (1977) · Zbl 0366.35029 [3] Kodama, Y.; Ablowitz, M.J., Perturbations of solitons and solitary waves, Stud. appl. math., 64, 225-245, (1981) · Zbl 0486.76029 [4] Olver, P.J., Hamiltonian perturbation theory and water waves, Contemporary mathematics, 28, 231-249, (1984) · Zbl 0521.76018 [5] Herman, R.L., A direct approach to studying soliton perturbations, J. phys. A, math. gen., 23, 2327-2362, (1990) · Zbl 0725.35088 [6] Tanaka, M., Perturbations on the K-dv equation—an approach based on the multiple time scale expansion, J. phys. soc. jpn, 49, 807-812, (1980) · Zbl 1334.35298 [7] Grimshaw, R.; Mitsudera, H., Slowly varying solitary wave solutions of the perturbed Korteweg-de Vries equation revisited, Stud. appl. math., 90, 75-86, (1993) · Zbl 0783.35062 [8] Elgin, J.N.; Brabec, T.; Kelly, S.M.J., A perturbative theory of soliton propagation in the presence of the third order dispersion, Opt. commun., 114, 321-328, (1995) [9] Ballantyne, G.J.; Gough, P.T.; Taylor, D.P., Deriving average soliton equations with a perturbation method, Phys. rev. E, 5, 825-828, (1995) [10] Currò, C.; Donato, A.; Povzner, A.Ya., Perturbation method for a generalized Burger’s equation, Int. J. non-linear mech., 27, 149-155, (1992) · Zbl 0825.35103 [11] Matsuno, Y., Multisoliton perturbation theory for the Benjamin-Ono equation and its application to real physical systems, Phys. rev. E, 51, 1471-1483, (1995) [12] Kivshar, Y.S.; Malomed, B.A., Dynamics of solitons in nearly integrable systems, Rev. mod. phys., 61, 763-915, (1989) [13] Tamizhmani, K.M.; Lakshmanan, M., Complete integrability of the Korteweg-de Vries equation under perturbation around its solution: Lie-Bäcklund symmetry approach, J. phys. A, math. gen., 16, 3773-3782, (1983) · Zbl 0544.35079 [14] Ma, W.X.; Fuchssteiner, B., The bi-Hamiltonian structure of the perturbation equations of KdV hierarchy, Phys. lett. A, 213, 49, (1995) · Zbl 1073.37537 [15] Kraenkel, R.A.; Manna, M.A.; Pereira, J.G., The Korteweg-de Vries hierarchy and long water-waves, J. math. phys., 36, 307-320, (1995) · Zbl 0826.76009 [16] Kraenkel, R.A.; Manna, M.A.; Montero, J.C.; Pereira, J.G., Boussinesq solitary-wave as a multiple-time solution of the Korteweg-de Vries hierarchy, J. math. phys., 36, 6822-6828, (1995) · Zbl 0844.35108 [17] Fuchssteiner, B.; Fokas, A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4, 47-66, (1981) · Zbl 1194.37114 [18] Fuchssteiner, B., Coupling of completely integrable systems: the perturbation bundle, (), 125-138 · Zbl 0786.35123 [19] Oevel, W., Dirac constraints in the field theory: lifts of Hamiltonian systems to the cotangent bundle, J. math. phys., 29, 210-219, (1988) · Zbl 0656.70011 [20] Ma, W.X., Poisson manifolds and classical Hamiltonian operators, Northeastern mathematical journal, 6, 346-356, (1990) · Zbl 0733.58017 [21] Fuchssteiner, B., Application of hereditary symmetries to nonlinear evolution equations, Nonlinear anal. theory methods appl., 3, 849-862, (1979) · Zbl 0419.35049 [22] Gel’fand, I.M.; Dorfman, I.Y., Hamiltonian operators and algebraic structures related to them, Funct. anal. appl., 13, 248-262, (1979) · Zbl 0437.58009 [23] Magri, F., A simple model of the integrable Hamiltonian equation, J. math. phys., 19, 1156-1162, (1978) · Zbl 0383.35065 [24] Chen, H.H.; Lin, J.E., On the integrability of multidimensional nonlinear evolution equations, J. math. phys., 28, 347-350, (1987) · Zbl 0663.35066 [25] Ma, W.X., Symmetry constraint of mkdv equations by binary nonlinearization, Physica A, 219, 467-481, (1995) [26] Wadati, M., The modified Korteweg-de Vries equation, J. phys. soc. jpn, 34, 1289-1296, (1973) · Zbl 1334.35299 [27] Ma, W.X., K-symmetries and τ-symmetries of evolution equations and their Lie algebras, J. phys. A, math. gen., 23, 2707-2716, (1990) · Zbl 0722.35078 [28] Ma, W.X., The algebraic structures of isospectral Lax operators and applications to integrable equations, J. phys. A, math. gen., 25, 5329-5343, (1992) · Zbl 0782.35072 [29] Zakharov, V.E.; Konopelchenko, B.G., On the theory of recursion operator, Commun. math. phys., 94, 483-509, (1984) · Zbl 0594.35080 [30] Dorfman, I.Ya.; Fokas, A.S., Hamiltonian theory over noncommutative rings and integrability in multidimensions, J. math. phys., 33, 2504-2514, (1992) · Zbl 0771.58025 [31] Schwarz, F., Symmetries of the two-dimensional Korteweg-de Vries equation, J. phys. soc. jpn, 51, 2387-2388, (1982) [32] David, D.; Kamran, N.; Levi, D.; Winternitz, P., Symmetry reduction for the Kadomtsev-Petviashvili equation using a loop algebra, J. math. phys., 27, 1225-1237, (1986) · Zbl 0598.35117 [33] Ma, W.X., The generators of vector fields and the time dependent symmetries of evolution equations, Sci. China A, 34, 769-782, (1991) · Zbl 0752.35062 [34] Fuchssteiner, B., Master symmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Prog. theor. phys., 70, 1508-1522, (1983) · Zbl 1098.37536
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