Stationary and 2+1 dimensional integrable reductions of AKNS hierarchy

Abstract

The main concepts of the soliton theory and infinite dimensional Hamiltonian Systems, including AKNS (Ablowitz, Kaup, Newell, Segur) integrable hierarchy of nonlinear evolution equations are introduced.By integro-differential recursion operator for this hierarchy, several reductions to KDV, MKdV, mixed KdV/MKdV and Reaction-Diffusion system are constructed.The stationary reduction of the fifth order KdV is related to finite-dimensional integrable system of Henon-Heiles type.Different integrable extensions of Henon-Heiles model are found with corresponding separation of variables in Hamilton-Jacobi theory.Using the second and the third members of AKNS hierarchy, new method to solve 2+1 dimensional Kadomtsev-Petviashvili(KP-II) equation is proposed.By the Hirota bilinear method, one and two soliton solutions of KP-II are constructed and the resonance character of their mutual interactions are studied.By our bilinear form we first time created new four virtual soliton resonance solution for KPII.Finally, relations of our two soliton solution with degenerate four soliton solution in canonical Hirota form of KPII are established.