Infinite hierarchy of nonlinear Schrödinger equations and their solutions

We study the infinite integrable nonlinear Schrödinger equation (NLSE) hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, AB breathers, Kuznetsov-Ma breathers, periodic solutions and rogue wave solutions for this infinite order hierarchy. We find that 'even' order equations in the set affect phase and 'stretching factors' in the solutions, while 'odd' order equations affect the velocities. Hence 'odd' order equation solutions can be real functions, while 'even' order equation solutions are always complex