Higher symmetries and exact solutions of linear and nonlinear Schrödinger equation

Abstract

A new approach for the analysis of partial differential equations is developed which is characterized by a simultaneous use of higher and conditional symmetries. Higher symmetries of the Schrödinger equation with an arbitrary potential are investigated. Nonlinear determining equations for potentials are solved using reductions to Weierstrass, Painlevé, and Riccati forms. Algebraic properties of higher order symmetry operators are analyzed. Combinations of higher and conditional symmetries are used to generate families of exact solutions of linear and nonlinear Schrödinger equations.

Heisenberg-type higher order symmetries of superintegrable systems separable in Cartesian coordinates

“…The one dimensional case of higher order symmetries has been studied in [2,3]. It is also worth to mention references [4][5][6][7][8], dealing with higher order symmetries, which are more related to our approach.…”

The nonlinear Dirac equation in Bose–Einstein condensates: Foundation and symmetries

“…Nonlinear Dirac equations have a long history in the literature, particularly in the context of particle and nuclear theory [12,13,14,15], but also in applied mathematics and nonlinear dynamics [16,17,18,19,20]. As nonlinearity is a ubiquitous aspect of Nature, it is natural to ask how nonlinearity might appear in a relativistic setting.…”

Nonlinear Dirac Equations

“…Other examples we hope to study with the nonlinear equations are CP violation and dark matter/energy. In this regard, it would be useful to obtain non-plane-wave solutions to our nonlinear equations, similar to what has been done for simpler polynomial-type nonlinear Dirac equations in [22,23].…”

“…Note that the multiparticle version of the above equation is separable, so that does not impose additional constraints. Nonlinear Dirac equations without derivatives in the nonlinear part have been studied in [22,23] and (3.3) is a special case of the equations studied there.…”

“…In [21] separability was imposed in a somewhat different manner from what we do here, but more importantly the authors of [21] only considered nonlinear Dirac equations that are obtained from the linear Dirac equation through a process of gauge-completion: thus their class of equations is more restrictive than ours. Some further contrasts of our procedure compared to others [22,23] is that we allow derivatives of the wavefunction in F , and also study nonlinearities which violate one or more of the discrete P, C, T symmetries as such cases are expected to be phenomenologically relevant.…”