Constructing new nonlinear evolution equations with supersymmetry

TL;DR: The ( G ′ / G , 1 / G ) -expansion method is applied to acquire some new, exact solutions of certain interesting, nonlinear, fractional-order partial differential equations arising in mathematical physics.

Abstract: In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable. In this lecture I review the theoretical formulation of supersymmetric quantum mechanics and discuss many of its applications. I show that the well-known exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials and shape invariance. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Further, it is pointed out that the connection between the solutions of the Dirac equation and the Schrödinger equation is exactly same as between the solutions of the MKdV and the KdV equations.

TL;DR: It is shown that supersymmetry can provide a versatile platform in synthesizing a new class of optical structures with desired properties and functionalities and that one-dimensional supersymmetric pairs display identical reflectivities and transmittivities for any angle of incidence in the Helmholtz regime.

Abstract: Exemption phaseout. You lose part of the benefit of your exemptions if your adjusted gross income (AGI) is above a certain amount. For 2009, the phaseout begins at $125,100 for married persons filing separately; $166,800 for single individuals; $208,500 for heads of household; and $250,200 for married persons filing jointly or qualifying widow(ers). However, in 2009, you can lose no more than 1/3 of the amount of your exemptions. In other words, each exemption cannot be reduced to less than $2,433.

TL;DR: This work presents the first experimental realization of optical supersymmetry and demonstrates mode conversion and global phase-matching between SUSY partner structures, which may pave the way for compact and highly efficient integrated mode-division-multiplexing schemes.