Integrability of nonlinear systems and perturbation theory.

*(English)*Zbl 0807.35107
What is integrability, Springer Ser. Nonlinear Dyn., 185-250 (1991).

[For the entire collection see Zbl 0724.00014.]

This article is concerned with the attempt to generalise to certain infinite-dimensional Hamiltonian systems results of PoincarĂ© and Birkhoff on the nonexistence of integrals of motion and reduction to normal form, respectively, appropriate to finite-dimensional mechanical systems. To do this the authors define, on a space dual to that of the independent spatial variables, a scattering operator by a limiting process from modified equations in which the nonlinear interaction terms are turned off asymptotically in time (it is unclear for what class of equations such a process is meaningful; certainly not for those with sufficiently strong singularities in their solutions). The terms in the formal series for the scattering operator have singularities on resonant manifolds; cf. the finite-dimensional case. Further, in order for an additional invariant of motion of a given form to exist it is necessary that on each resonant manifold either the corresponding coefficient of the scattering operator vanish or certain relations between sets of coefficients of the additional invariant hold, in which case the dispersion laws are said to be degenerate. The authors analyse the structure of such degenerate dispersion laws. An important statement is that for a system with nondegenerate dispersion laws the existence of one additional invariant is enough to guarantee the existence of an infinite number of invariants. So far the theory is restricted to solution classes of rapid decay. For periodic problems the authors discuss canonical transformation to normal form. The existence of an additional integral apparently obviates the problem of small denominators. The Kadomtsev- Petviashvili-II and Veselov-Novikov equations have nondegenerate dispersion laws, the Kadomtsev-Petviashvili-I equation a degenerate dispersion law. This latter equation is not integrable. Finally, the Davey-Stewartson equation and application to one spatial dimension are discussed.

This article is concerned with the attempt to generalise to certain infinite-dimensional Hamiltonian systems results of PoincarĂ© and Birkhoff on the nonexistence of integrals of motion and reduction to normal form, respectively, appropriate to finite-dimensional mechanical systems. To do this the authors define, on a space dual to that of the independent spatial variables, a scattering operator by a limiting process from modified equations in which the nonlinear interaction terms are turned off asymptotically in time (it is unclear for what class of equations such a process is meaningful; certainly not for those with sufficiently strong singularities in their solutions). The terms in the formal series for the scattering operator have singularities on resonant manifolds; cf. the finite-dimensional case. Further, in order for an additional invariant of motion of a given form to exist it is necessary that on each resonant manifold either the corresponding coefficient of the scattering operator vanish or certain relations between sets of coefficients of the additional invariant hold, in which case the dispersion laws are said to be degenerate. The authors analyse the structure of such degenerate dispersion laws. An important statement is that for a system with nondegenerate dispersion laws the existence of one additional invariant is enough to guarantee the existence of an infinite number of invariants. So far the theory is restricted to solution classes of rapid decay. For periodic problems the authors discuss canonical transformation to normal form. The existence of an additional integral apparently obviates the problem of small denominators. The Kadomtsev- Petviashvili-II and Veselov-Novikov equations have nondegenerate dispersion laws, the Kadomtsev-Petviashvili-I equation a degenerate dispersion law. This latter equation is not integrable. Finally, the Davey-Stewartson equation and application to one spatial dimension are discussed.

##### MSC:

35Qxx | Partial differential equations of mathematical physics and other areas of application |

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |