Articles written in Pramana – Journal of Physics
Volume 84 Issue 3 March 2015 pp 339-352
Different types of breathers and rogue waves (RWs) are some of the important coherent structures which have been recently realized in several physical phenomena in hydrodynamics, nonlinear optics, Bose–Einstein condensates, etc. Mathematically, they have been deduced in non-linear Schrödinger (NLS) equations. Here we show the existence of general breathers, Akhmediev breathers, Ma soliton and rogue wave solutions in coupled Manakov-type NLS equations and coupled generalized NLS equations representing four-wave mixing. We deduce their explicit forms using Hirota bilinearization procedure and bring out their exact structures and important properties. We also show the method to deduce the various breather solutions from rogue wave solutions using factorization form and the so-called imbricate series.
Volume 85 Issue 5 November 2015 pp 755-787
Lie symmetry analysis is one of the powerful tools to analyse nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries, contact symmetries, hidden symmetries, nonlocal symmetries, 𝜆-symmetries, adjoint symmetries and telescopic vector fields of a secondorder ordinary differential equation. We also illustrate the algorithm involved in each method by considering a nonlinear oscillator equation as an example. The connections between
symmetries and integrating factors and
symmetries and integrals are also discussed and illustrated through the same example.
The interconnections between some of the above symmetries, i.e.,
Lie point symmetries and 𝜆-symmetries and
exponential nonlocal symmetries and 𝜆-symmetries are also discussed.
The order reduction procedure is invoked to derive the general solution of the second-order equation.
Volume 96, 2022
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