Analysis and comparative study of non-holonomic and quasi-integrable deformations of the nonlinear Schrödinger equation

The non-holonomic deformation of the nonlinear Schrödinger equation, uniquely obtained from both the Lax pair and Kupershmidt’s bi-Hamiltonian (Kupershmidt in Phys Lett A 372:2634, 2008) approaches, is compared with the quasi-integrable deformation of the same system (Ferreira et al. in JHEP 2012:103, 2012). It is found that these two deformations can locally coincide only when the phase of the corresponding solution is discontinuous in space, following a definite phase-modulus coupling of the non-holonomic inhomogeneity function. These two deformations are further found to be not gauge equivalent in general, following the Lax formalism of the nonlinear Schrödinger equation. However, the localized solutions corresponding to both these cases converge asymptotically as expected. Similar conditional correspondence of non-holonomic deformation with a non-integrable deformation, namely due to locally scaled amplitude of the solution to the nonlinear Schrödinger equation, is further obtained.