TY - JOUR

T1 - Travelling-wave and separated variable solutions of a nonlinear Schroedinger equation

AU - Bountis, Tassos

AU - Nobre, Fernando D.

N1 - Funding Information:
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).

PY - 2016/8/1

Y1 - 2016/8/1

N2 - 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 Ψ(x, t), 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 Φ(x, t), derived recently within the context of a classical field theory, which corresponds to Ψ*(x, t) 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 < 8. However, we also identify ranges of q values for which these solutions vanish at infinity, and may therefore be physically important.

AB - 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 Ψ(x, t), 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 Φ(x, t), derived recently within the context of a classical field theory, which corresponds to Ψ*(x, t) 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 < 8. 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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U2 - 10.1063/1.4960723

DO - 10.1063/1.4960723

M3 - Article

AN - SCOPUS:84983285037

VL - 57

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 8

M1 - 082106

ER -