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Travelling-wave and separated variable solutions of a nonlinear Schroedinger equation

### 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 q and reduce to the corresponding standard linear equations in the limit q → 1. Their main virtue is that they possess plane-wave solutions expressed in terms of a q-exponential function that can vanish at infinity, while preserving the Einstein energy-momentum relation for all q. 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 q-exponential framework, and examine their behavior at infinity for different values of q. 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 q → 1. For x ∈ ℜ, we show that certain perturbations of these q-exponential solutions Ψ(x, t) and Φ(x, t) are unbounded and hence would lead to divergent probability densities over the full domain −∞ < x < ∞. However, we also identify ranges of q values for which these solutions vanish at infinity, and may therefore be physically important.

Published by AIP Publishing.

Received 22 March 2016
Accepted 29 July 2016
Published online 17 August 2016

Acknowledgments:
We acknowledge many useful conversations with Professor C. Tsallis. One of us (T.B.) is grateful for the support provided by the research project MACOMSYS co-financed by the European Union (European Social Fund = ESF) and Greek National Funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES. Investing in knowledge society through the European Social Fund. F.D.N. thanks partial financial support from CNPq and FAPERJ (Brazilian funding agencies).

Article outline:

I. INTRODUCTION
II. TRAVELLING-WAVE SOLUTIONS
A. Wave solutions of the NLSE
B. Travelling wave solutions of the associated NLSE equation
III. SEPARATED VARIABLE SOLUTIONS OF THE NLSE
A. Solutions of the principal NLSE
B. Separated variable solutions of the associated equation
C. Probability density for a particular solution
IV. CONTINUITY EQUATION
V. CONCLUSIONS

/content/aip/journal/jmp/57/8/10.1063/1.4960723

1.

R. L. Liboff, Introductory Quantum Mehanics, 4th ed. (Addison Wesley, San Francisco, 2003).

2.

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, New York, 1999).

3.

A. C. Scott, The Nonlinear Universe (Springer, Berlin, 2007).

9.

C. Tsallis, Introduction to Nonextensive Statistical Mechanics (Springer, New York, 2009).

17.

Herein we adopt the notation of Refs. 5 and 6, where the fields and represent dimensionless quantities, rescaled by their corresponding amplitudes.

http://aip.metastore.ingenta.com/content/aip/journal/jmp/57/8/10.1063/1.4960723

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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 q and reduce to the corresponding standard linear equations in the limit q → 1. Their main virtue is that they possess plane-wave solutions expressed in terms of a q-exponential function that can vanish at infinity, while preserving the Einstein energy-momentum relation for all q. 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 q-exponential framework, and examine their behavior at infinity for different values of q. 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 q → 1. For x ∈ ℜ, we show that certain perturbations of these q-exponential solutions Ψ(x, t) and Φ(x, t) are unbounded and hence would lead to divergent probability densities over the full domain −∞ < x < ∞. However, we also identify ranges of q values for which these solutions vanish at infinity, and may therefore be physically important.

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