Integrable Boundary Conditions for the Nonlinear Schrödinger Hierarchy Via the Fokas Method

We study integrable boundary conditions for the whole hierarchy of nonlinear Schrödinger (NLS) equations defined on the half‐line. We find that the even‐order and odd‐order NLS equations admit rather different integrable boundary conditions. In particular, the odd‐order NLS equations admit a new class of integrable boundary conditions that involves time reversal. We establish the integrability of the NLS hierarchy with our new boundary conditions by demonstrating the existence of infinitely many integrals of motion in involution. Moreover, we develop the boundary dressing technique to construct soliton solutions satisfying these new boundary conditions.