Intertwining relations and Darboux transformations for the wave equations

Abstract

Using the intertwining operator technique we construct Darboux transformations for the wave equation with position dependent effective mass and with linearly energy dependent potentials. The for mally adjoint generators of supersymmetry and two formally self adjoint superpartner Hamiltonians are con structed and they close a quadratic pseudo superalgebra. The Darboux transformations are constructed in differential and integral forms and an interrelation is established between them. The approach is applied to generation of isospectral potentials with additional or removal bound states or construction of new partner potentials without changing the spectrum, i.e. fully isospectral potentials. The method is illustrated by some concrete examples. The influence of distance between levels on the form of potentials is investigated. In par ticular, asymmetric double well potentials are generated.

Supersymmetry-Inspired Non-Hermitian Optical Couplers

“…First, we note that the SUSY transformation in Eq. (10) is only a particular case of a more general class of Darboux-type transformations 45 , and that more general -symmetric profiles (e.g., with more than one gain/loss oscillation) for the original waveguide may be considered. It is therefore possible to generate classes of alternative isospectral (apart from a selected mode) partner waveguides whose coupling effects might be compatible with the desired functionality.…”

“…With specific reference to a few-mode scenario, we have shown that this phenomenon may be controlled to a certain extent by resorting to modified-SUSY partner profiles characterized by suitable spatial scaling and/or rearrangement of the gain and loss regions. For more complicated configurations requiring the selection of a particularly high-order mode, and hence the tailoring of the complex propagation constant of larger sets of supermodes, further degrees of freedom may be obtained via more complex -symmetric profiles as well as more general transformations 45 . Within this framework, of particular interest also appear some recently introduced architectures based on optical coupled networks amenable to a discretized Hamiltonian formulation 9 .…”

Dynamical features and sensitivity visualization of thin-film Polarisation equation

“…It is demonstrated that this equation has periodic soliton solutions. Investigating the solution of such nonlinear equations using other techniques, such as inverse scattering, transform [18], Darboux transform [19], and Tanh method, to mention a few, is important to offer the richness of the solution. Many new recent methods and works can be seen in these [20][21][22][23][24][25][26].…”