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Model equation of the theory of solitons. (English. Russian original) Zbl 1138.37044
Theor. Math. Phys. 153, No. 1, 1373-1387 (2007); translation from Teor. Mat. Fiz. 153, No. 1, 29-45 (2007).
Summary: We consider the hierarchy of integrable $$(1+2)$$-dimensional equations related to the Lie algebra of vector fields on the line. We construct solutions in quadratures that contain $$n$$ arbitrary functions of a single argument. A simple equation for the generating function of the hierarchy, which determines the dynamics in negative times and finds applications to second-order spectral problems, is of main interest. Considering its polynomial solutions under the condition that the corresponding potential is regular allows to develop a rather general theory of integrable $$(1+1)$$-dimensional equations.

##### MSC:
 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures 35Q58 Other completely integrable PDE (MSC2000) 35Q51 Soliton equations
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