The Eckhaus PDE

The evolution equation of nonlinear Schrodinger type i psi t+ psi xx+2( mod psi mod 2)x psi + mod psi mod 4 psi =0 psi identical to psi (x,t) is integrable; it can be (exactly) linearised by a change of (dependent) variable. Hence, many solutions of this PDE can be exhibited; several of them reveal a remarkable phenomenology, which, if one focuses on the evolution of the x derivative of the (squared) modulus of psi (x,t), may be described in terms of localised entities ('solitons' of various kinds, including 'breathers', 'boomerons' etc.) and of various reactions between them (decay of a soliton into two or more, and vice versa; inelastic collisions of two or more solitons etc.). The Cauchy problem is also solved, for initial data whose modulus tends (appropriately fast) to finite values at both ends, mod psi (+or- infinity ,0) mod 2=p+or-; it turns out that solitons are present if p+>0, that their phenomenology depends sensitively on the behaviour of psi (x,0) as x to + infinity and that a generic initial datum psi (x,0) yields solitons that only collide elastically (the same solitons are present in the remote past and future and they move with opposite velocities).