Lie algebra and equations of Korteweg-de Vries type. (Russian) Zbl 0558.58027
Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 81-180 (1984).
This review is mainly devoted to detailed description of the authors’ results concerning the construction of the integrable evolution equations of Korteweg-de Vries (KdV) type and their relation with Lie algebras and Kac-Moody algebras. Some general algebraic problems connected with the inverse scattering transform method are also considered. The local conservation laws, symmetries and Hamiltonian formalism are mainly discussed. This review contains complete proofs of all propositions and theorems.
In section 1 the methods which are used in the paper are demonstrated in the well-known case of \(N\)-interacting waves equations. Proposition 1.2., which is an algebraic version of the dressing method, plays an important role in all further constructions.
In section 2 the well-known results concerning the scalar Lax equation \(\partial L/\partial t=AL-LA\) (\(L\) and \(A\) are scalar differential operators) are described from the point of view of the theory of fractional degrees. It turns out that this equation is the KdV type equation connected with Kac-Moody algebra \(\mathfrak{sl}(k,C[\lambda,\lambda^{-1}])\), where \(k\) is the order of \(L\).
Section 3 is devoted to the detailed description of the relation between the scalar Lax equation and the Kac-Moody algebra \(\mathfrak{sl}(k,C[\lambda,\lambda^{-1}])\). The consideration of the gauge freedom for the spectral problems allows one to construct the KdV type equations for any Kac-Moody algebra. This approach also gives the natural group-theoretical treatment of the second Hamiltonian structure. In section 3 the analogues of mKdV equation and Miura transformation for the algebra \(\mathfrak{sl}(k,C[\lambda,\lambda^{-1}])\) are also constructed.
Section 4 contains the generalization of the results of section 1 to the case of the operator \(L=d/dx+\lambda a+q(x)\) where \(a\) and \(q(x)\) belong to an arbitrary Lie algebra.
In section 5 an elementary introduction to the theory of Kac-Moody algebras is presented. The properties of Kac-Moody algebras described here are used in the next sections. In section 6 the analogues of KdV and mKdV equations for arbitrary Kac-Moody algebras are defined. It occurs that one series of mKdV type equations and several series of KdV type equations correspond to each Kac-Moody algebra.
Section 7 is devoted to the construction of the scalar Lax pairs for KdV type equations corresponding to Kac-Moody algebras which are different from \(\mathfrak{sl}(k,C[\lambda,\lambda^{-1}])\). In section 8 the Hamiltonian formalism of the KdV type equations is considered. Section 9 contains some concrete examples of KdV and mKdV type equations. In section 10 the two-dimensional Toda lattice equations and their relation with the KdV type equations are considered.
In section 1 the methods which are used in the paper are demonstrated in the well-known case of \(N\)-interacting waves equations. Proposition 1.2., which is an algebraic version of the dressing method, plays an important role in all further constructions.
In section 2 the well-known results concerning the scalar Lax equation \(\partial L/\partial t=AL-LA\) (\(L\) and \(A\) are scalar differential operators) are described from the point of view of the theory of fractional degrees. It turns out that this equation is the KdV type equation connected with Kac-Moody algebra \(\mathfrak{sl}(k,C[\lambda,\lambda^{-1}])\), where \(k\) is the order of \(L\).
Section 3 is devoted to the detailed description of the relation between the scalar Lax equation and the Kac-Moody algebra \(\mathfrak{sl}(k,C[\lambda,\lambda^{-1}])\). The consideration of the gauge freedom for the spectral problems allows one to construct the KdV type equations for any Kac-Moody algebra. This approach also gives the natural group-theoretical treatment of the second Hamiltonian structure. In section 3 the analogues of mKdV equation and Miura transformation for the algebra \(\mathfrak{sl}(k,C[\lambda,\lambda^{-1}])\) are also constructed.
Section 4 contains the generalization of the results of section 1 to the case of the operator \(L=d/dx+\lambda a+q(x)\) where \(a\) and \(q(x)\) belong to an arbitrary Lie algebra.
In section 5 an elementary introduction to the theory of Kac-Moody algebras is presented. The properties of Kac-Moody algebras described here are used in the next sections. In section 6 the analogues of KdV and mKdV equations for arbitrary Kac-Moody algebras are defined. It occurs that one series of mKdV type equations and several series of KdV type equations correspond to each Kac-Moody algebra.
Section 7 is devoted to the construction of the scalar Lax pairs for KdV type equations corresponding to Kac-Moody algebras which are different from \(\mathfrak{sl}(k,C[\lambda,\lambda^{-1}])\). In section 8 the Hamiltonian formalism of the KdV type equations is considered. Section 9 contains some concrete examples of KdV and mKdV type equations. In section 10 the two-dimensional Toda lattice equations and their relation with the KdV type equations are considered.
Reviewer: B. G. Konopelchenko (Novosibirsk)
MSC:
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
| 37K15 | Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 37K60 | Lattice dynamics; integrable lattice equations |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |
