Integrable inhomogeneous NLS equations are equivalent to the standard NLS

A class of inhomogeneous nonlinear Schrödinger equations (NLS), claiming to be novel integrable systems with rich properties continues appearing in PhysRev and PRL. All such equations are shown to be not new but equivalent to the standard NLS, which trivially explains their integrability features. PACS no: 02.30.Ik , 04.20.Jb , 05.45.Yv , 02.30.Jr Time and again various forms of inhomogeneous nonlinear Schrödinger equations (IHNLS) along with their discrete variants are appearing as central result mostly in the pages of Phys. Rev, and PRL CLU76,RB87,PRA91,Kon93,PRL00,PRL05,PRL07 ̧ , which are either suspected to be integrable due to the finding of particular analytic or stable computer solutions, or assumed to be only Painlevé integrable arxiv08 ̧ , or else claimed to be completely new integrable systems. Apparently the solution of such integrable systems needs generalization of the inverse scattering method (ISM), in which the usual isospectral approach involving only constant spectral parameter λ has to be extended to nonisospectral flow with time-dependent λ(t). Moreover certain features of the soliton solutions of such inhomogeneous NLS, like the changing of the solitonic amplitude, shape and velocity with time were thought to be new and surprising discovery. We show here that all these IHNLS , though completely integrable are not new or independent integrable systems, and in fact are equivalent to the standard homogeneous NLS, linked through simple gauge, scaling and coordinate transformations. The standard NLS is a well known integrable system with known Lax pair, soliton solutions and usual isospectral ISM nls,ALM ̧ . As we see below, a simple time-dependent gauge transformation of the standard isospectral system with constant λ can create the illusion of having complicated nonisospectrality. Similarly, a time-dependent scaling of the standard NLS field Q → q = ρ(t)Q would naturally lead the constant soliton amplitude to a time-dependent one. In the same way a trivial coordinate transformation x → X = ρ(t)x would change the usual constant velocity v of the NLS soliton to a time-variable quantity v(t) = v ρ(t) and the invariant shape of the standard soliton with constant extension Γ = 1 κ to a time-dependent one with variable extension Γ(t) = Γ ρ(t) (see Fig 1a a,b). Therefore all the rich integrability properties of the IHNLS, observed in earlier papers, including more exotic and seemingly surprising features like nonisospectral flow, appearance of shape changing and accelerating soliton etc. can be trivially explained from the timedependent transformations of these IHNLS from the standard NLS and the corresponding explicit result , namely the Lax pair, N-soliton solutions, infinite conserved quantities etc. for the inhomogeneous NLS models can be derived easily from their well known counterparts in the homogeneous NLS case through the same transformations nls ̧ .