Nonlinear Gauge Transformations and Exact Solutions of the Doebner-Goldin Equation

We describe the behaviour of semiclassical electrodynamics under gauge transformations. For this purpose we observe the structure of Schr\"odinger equation and matricial elements under these transformations. We conclude this theory is not gauge invariant. As a consequence of this fact, we obtain a possible loss of predictability of physical results.

We harness the freedom in the celebrated gauge transformation approach to generate dark solitons of coupled nonlinear Schr\"odinger (NLS) type equations. The new approach which is purely algebraic could prove to be very useful, particularly in the construction of vector dark solitons in the fields of nonlinear optics, plasma physics and Bose-Einstein condensates. We have employed this algebraic method to coupled Gross- Pitaevskii (GP) and NLS equations and obtained dark solit...

The Schr\"{o}dinger equation of a charged particle in a uniform electric field can be specified in either a time-independent or a time-dependent gauge. The wave-function solutions in these two gauges are related by a phase-factor reflecting the gauge symmetry of the problem. In this article we show that the effect of such a gauge transformation connecting the two wave-functions can be mimicked by the effect of two successive extended Galilean transformations connecting the tw...

Due to its connection to the diffeomorphism group, nonlinear quantum mechanics may play an important role in quantum geometry. The Doebner-Goldin nonlinearity (arising from representations of the diffeomorphism group) amplifies nonlocal signaling effects under extreme localization, suggesting that even if greatly suppressed at low energies, such effects may be significant at the Planck scale. This offers new perspectives on Planck-scale physics.

We consider Darboux transformations for the derivative nonlinear Schr\"odinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, the parabolic and soliton solutions of the derivative nonlinear Schr\"odinger equation are given as explicit examples.

We introduce N-parameter perturbation theory as a new tool for the study of non-linear relativistic phenomena. The main ingredient in this formulation is the use of the Baker-Campbell-Hausdorff formula. The associated machinery allows us to prove the main results concerning the consistency of the scheme to any perturbative order. Gauge transformations and conditions for gauge invariance at any required order can then be derived from a generating exponential formula via a simp...

In the recent comment quoted in the title (arXiv:1407.7852v1), a comment is presented on our recent work which derive a generalized nonlinear wave solution formula for mixed coupled nonlinear Sch\"{o}dinger equations by performing the unified Darboux transformation (arXiv:1407.5194). Here we would like to reply to the comment and clarify some facts in arXiv:1407.5194.

We consider the kinetic derivative nonlinear Schr\"odinger equation, which is a one-dimensional nonlinear Schr\"odinger equation with a cubic derivative nonlinear term containing the Hilbert transformation. In our previous work, we proved small-data global well-posedness of the Cauchy problem on the torus in Sobolev space $H^s$ for $s>1/2$ by combining the Fourier restriction norm method with the parabolic smoothing effect, which is available in the periodic setting. In this ...

In classical mechanics, we can describe the dynamics of a given system using either the Lagrangian formalism or the Hamiltonian formalism, the choice of either one being determined by whether one wants to deal with a second degree differential equation or a pair of first degree ones. For the former approach, we know that the Euler-Lagrange equation of motion remains invariant under additive total derivative with respect to time of any function of coordinates and time in the L...

By using the multipolar gauge it is shown that the quantum mechanics of an electrically charged particle moving in a prescribed classical electromagnetic field (wave mechanics) may be expressed in a manner that is gauge invariant, except that the only gauge functions that are allowable in a gauge transformation are those that consist of the sum of a function that depends only on the space coordinates and a term that is the product of a constant and the time coordinate. The mu...