2013/11/18 by Partha Guha, Indranil Mukherjee, Guha, Partha +1
Mathematics · Physics and Astronomy · #14G32 #35Q53 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Fractional Differential Equations Solutions #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.1311.4334
openalex publication_date 2013/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The objective of this work is to explore the class of equations of the\nNon-linear Schrodinger type by employing the Adler-Kostant-Symes theorem and\nthe Tu methodology.In the first part of the work, the AKS theory is discussed\nin detail showing how to obtain the non-linear equations starting from a\nsuitably chosen spectral problem.Equations derived by this method include\ndifferent members of the NLS family like the NLS, the coupled KdV type NLS, the\ngeneralized NLS, the vector NLS, the Derivative NLS, the Chen-Lee-Liu and the\nKundu-Eckhaus equations. In the second part of the paper, the steps in the Tu\nmethodology that are used to formulate the hierarchy of non-linear evolution\nequations starting from a spectral problem, are outlined. The AKNS,\nKaup-Newell, and generalized DNLS hierarchies are obtained by using this\nalgorithm. Several reductions of the hierarchies are illustrated. The famous\ntrace identity is then applied to obtain the Hamiltonian structure of these\nhierarchies and establish their complete integrability. In the last part of the\npaper, the non-holonomic deformation of the class of integrable systems\nbelonging to the NLS family is studied. Equations examined include the NLS,\ncoupled KdV-type NLS and Derivative NLS (both Kaup-Newell and Chen-Lee-Liu\nequations). NHD is also applied to the hierarchy of equations in the AKNS\nsystem and the KN system obtained through application of the Tu\nmethodology.Finally, we discuss the connection between the two formalisms and\nindicate the directions of our future endeavour in this area.\n