Analysis and comparative study of non-holonomic and quasi-integrable deformations of the nonlinear Schrödinger equation
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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.
KeywordsNonlinear Schrödinger equation Quasi-integrable deformation Non-holonomic deformation Solitons
The authors are grateful to Professors Luiz. A. Ferreira and Wojtek J. Zakrzewski for their encouragement, various useful discussions and critical reading of the draft.
Kumar Abhinav’s research is done at and supported by the IF, Naresuan University, Thailand. Partha Guha’s research is partially supported by FAPESP through Grant numbered 2016/06560-6. A pre-print of this manuscript is available online at arXiv:1611.00961 [nlin.SI].
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Conflict of interest
The authors declare that they have no conflict of interest concerning the publication of this manuscript.
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