Finite-dimensional reductions of conservative dynamical systems and numerical analysis. I (Q2768825)

This paper deals with bi-Hamiltonian infinite dimensional dynamical systems of the form \(u_t=K[u], K:M \mapsto T(M)\), determined on the functional manifold \(M\) of smooth \(2\pi \)-periodic functions. The goal of the authors is to find various types of solutions (such as travelling waves, solitons, etc.) by employing the method of finite-dimensional reductions to invariant submanifolds of the phase space. NEWLINENEWLINENEWLINEFirst the authors recall a construction of finite dimensional Hamiltonian systems generated by vector fields \(d/dt\) and \(d/dx\) on the manifold \(M_N\) of fixed points of dynamical system with Hamiltonian \(H_N+\sum_{j=0}^{N-1}c_jH_j\) where \(\{H_i\}\) is a hierarchy of conserved quantities. This construction is then applied to KdV equation, modified nonlinear Schrödinger equation, and a nonlinear hydrodynamical system. Time evolution of initial data lying on \(M_N\) is studied with the help of qualitative and numerical methods.