ar X iv : s ol v - in t / 9 70 70 08 v 1 1 4 Ju l 1 99 7 Functional representation of the Ablowitz - Ladik hierarchy .

ar X iv : s ol v - in t / 9 40 70 05 v 1 2 7 Ju l 1 99 4 Integrable dynamics of a discrete curve and

We show that the following elementary geometric properties of the motion of a discrete (i.e. piecewise linear) curve select the integrable dynamics of the Ablowitz-Ladik hierarchy of evolution equations: i) the set of points describing the discrete curve lie in the sphere S; ii) the distance between any two subsequent points does not vary in time; iii) the dynamics does not depend explicitly on...

ar X iv : s ol v - in t / 9 70 70 14 v 1 2 7 Ju l 1 99 7 The constrained modified KP hierarchy and the generalized Miura transformations

In this letter, we consider the second Hamiltonian structure of the constrained modified KP hierarchy. After mapping the Lax operator to a pure differential operator the second structure becomes the sum of the second and the third Gelfand-Dickey brackets defined by this differential operator. We simplify this Hamiltonian structure by factorizing the Lax operator into linear terms.

ar X iv : s ol v - in t / 9 60 70 03 v 1 1 6 Ju l 1 99 6 Coupled Integrable Systems Associated with a Polynomial Spectral Problem and their Virasoro Symmetry

An isospectral hierarchy of commutative integrable systems associated with a polynomial spectral problem is proposed. The resulting hierarchy possesses a recursion structure controlled by a hereditary operator. The nonisospectral flows generate the time first order dependent symmetries of the isospectral hierarchy, which constitute Virasoro symmetry algebras together with commutative symmetries.

ar X iv : s ol v - in t / 9 40 70 03 v 1 1 8 Ju l 1 99 3 INVARIANT INTEGRABILITY CRITERION FOR THE EQUATIONS OF HYDRODYNAMICAL TYPE

Invariant integrability criterion for the equations of hydrodynamical type is found. This criterion is written in the form of vanishing for some tensor which is derived from the velocities matrix of hydrodynamical equations.

ar X iv : s ol v - in t / 9 70 80 06 v 1 1 4 A ug 1 99 7 Introduction to the Hirota bilinear method