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New hierarchy of symmetries. Lie groups and the soliton equations in the nonisospectral case
3rd workshop on nonlinear evolution equations
227
NONLINEAR EQUATIONS AND OPERATOR ALGEBRAS
V. A. Marchenko, Institute for Low Temperature Sciences, Kharkov, USSR
Physics and Engineering,
Ukr. SSR Academy of
The lecture treats the method of integrating equations of Korteweg-de Vries type. The method is based on the substitution of the given equation by another one of the same form with respect to operator-valued functions taking values in an arbitrary operator algebra. One-soliton solutions (i.e. solutions in the form of a propagating wave) of the operator equation are found without difficulty, and solutions of the initial equation are obtained from them by bordering with finitedimensional constant projectors. Various classes of solutions are obtained by varying the operator algebra; beside already known solutions, others are obtained, not reducible to the former. Realization of this approach in concrete algebras and investigation of the solutions obtained are related to some classical problems of calculus and their generalizations.
ANALYTIC
INTEGRABILITY
M. D. Kruskal, Mathematics Department,
Princeton University,
Princeton,
NJ 08544, USA
It is proposed that the so-called "Painleve'test" for integrability of an analytic ordinary differential equation (o.d.e.) viz. that all movable singularities of its solutions in the complex plane be (no worse than) poles, be interpreted as a test for the single-valuedness of integrals or solutions. The test is surprisingly good in spite of several anomalies and mysteries, which can be largely resolved by recasting it as a "polyPainleve" test, in which several singularities are considered simultaneously. The danger is multi-branching or multi-valuedness, not discrete but densely distributed. A legitimate integral is defined accordingly, and can very often be tested for by appropriate singular perturbation expansion.
NEW HIERARCHY OF SYMMETRIES. NONISOSPECTRAL CASE Y. S. Li, G. C. Zhu, Department of Mathematics, Anhui, China
LIE GROUPS AND THE SOLITON EQUATIONS
University
of Science
and Technology,
IN THE
Hefei,
We point out the relation between the new symmetries for some integrable equations and nonisospectral evolution equations and prove that the new set of symmetries with the old set of symmetries construct an infinite dimensional Lie algebra.
227
NONLINEAR EQUATIONS AND OPERATOR ALGEBRAS
V. A. Marchenko, Institute for Low Temperature Sciences, Kharkov, USSR
Physics and Engineering,
Ukr. SSR Academy of
The lecture treats the method of integrating equations of Korteweg-de Vries type. The method is based on the substitution of the given equation by another one of the same form with respect to operator-valued functions taking values in an arbitrary operator algebra. One-soliton solutions (i.e. solutions in the form of a propagating wave) of the operator equation are found without difficulty, and solutions of the initial equation are obtained from them by bordering with finitedimensional constant projectors. Various classes of solutions are obtained by varying the operator algebra; beside already known solutions, others are obtained, not reducible to the former. Realization of this approach in concrete algebras and investigation of the solutions obtained are related to some classical problems of calculus and their generalizations.
ANALYTIC
INTEGRABILITY
M. D. Kruskal, Mathematics Department,
Princeton University,
Princeton,
NJ 08544, USA
It is proposed that the so-called "Painleve'test" for integrability of an analytic ordinary differential equation (o.d.e.) viz. that all movable singularities of its solutions in the complex plane be (no worse than) poles, be interpreted as a test for the single-valuedness of integrals or solutions. The test is surprisingly good in spite of several anomalies and mysteries, which can be largely resolved by recasting it as a "polyPainleve" test, in which several singularities are considered simultaneously. The danger is multi-branching or multi-valuedness, not discrete but densely distributed. A legitimate integral is defined accordingly, and can very often be tested for by appropriate singular perturbation expansion.
NEW HIERARCHY OF SYMMETRIES. NONISOSPECTRAL CASE Y. S. Li, G. C. Zhu, Department of Mathematics, Anhui, China
LIE GROUPS AND THE SOLITON EQUATIONS
University
of Science
and Technology,
IN THE
Hefei,
We point out the relation between the new symmetries for some integrable equations and nonisospectral evolution equations and prove that the new set of symmetries with the old set of symmetries construct an infinite dimensional Lie algebra.