Algebraic geometry and stability for integrable systems

Abstract

In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms of theta functions of Riemann surfaces. However, the explicit formulas obtained in this way fail to answer qualitative questions such as whether a given singular solution is stable or not. In the present paper, the problem of stability for equilibrium points is considered, and it is shown that this problem can also be approached by means of algebraic geometry.

References in corpus (5)

• Stability of equilibria for the $\mathfrak{so}(4)$ free rigid body
• The Jacobi-Rosochatius problem on an ellipsoid: the Lax representations and billiards
• Stability in bi-Hamiltonian systems and multidimensional rigid body
• On the stability problem for the $\mathfrak{so}(5)$ free rigid body
• A note on relative equilibria of multidimensional rigid body

Cited by in corpus (2)

• Stability in bi-Hamiltonian systems and multidimensional rigid body
• Singularities of integrable systems and nodal curves