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/content/aip/journal/jmp/57/8/10.1063/1.4960723
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Herein we adopt the notation of Refs. 5 and 6, where the fields and represent dimensionless quantities, rescaled by their corresponding amplitudes.
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/content/aip/journal/jmp/57/8/10.1063/1.4960723
2016-08-17
2016-09-29

Abstract

Some interesting nonlinear generalizations have been proposed recently for the linear Schroedinger, Klein-Gordon, and Dirac equations of quantum and relativistic physics. These novel equations involve a real parameter and reduce to the corresponding standard linear equations in the limit → 1. Their main virtue is that they possess plane-wave solutions expressed in terms of a -exponential function that can vanish at infinity, while preserving the Einstein energy-momentum relation for all . In this paper, we first present new travelling wave and separated variable solutions for the main field variable , of the nonlinear Schroedinger equation (NLSE), within the -exponential framework, and examine their behavior at infinity for different values of . We also solve the associated equation for the second field variable , derived recently within the context of a classical field theory, which corresponds to for the linear Schroedinger equation in the limit → 1. For ∈ ℜ, we show that certain perturbations of these -exponential solutions Ψ(, ) and Φ(, ) are unbounded and hence would lead to divergent probability densities over the full domain −∞ < < ∞. However, we also identify ranges of values for which these solutions vanish at infinity, and may therefore be physically important.

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