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Solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity.
Fred Cooper1, Avinash Khare, Bogdan Mihaila
1Santa Fe Institute, Santa Fe, New Mexico 87501, USA. fcooper@lanl.gov
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
Summary
We found exact analytic solitary waves for nonlinear Dirac equations (NLDEs) with scalar and vector self-interactions. These generalize nonlinear Schrödinger equation (NLSE) solutions and reveal a 1/2m correction in the nonrelativistic limit.
Area of Science:
- Theoretical Physics
- Quantum Field Theory
- Nonlinear Dynamics
Background:
- Nonlinear Dirac equations (NLDEs) describe relativistic quantum systems.
- Exact solutions for NLDEs are crucial for understanding solitary wave behavior.
- Generalizing known solutions like those for the nonlinear Schrödinger equation (NLSE) is of significant interest.
Purpose of the Study:
- To find exact analytic forms for solitary waves in 1+1 dimensional NLDEs with specific scalar-scalar and vector-vector self-interactions.
- To establish the relationship between these NLDE solitary waves and NLSE solutions.
- To investigate the nonrelativistic limit and stability of these solitary waves.
Main Methods:
- Solving the NLDEs with scalar-scalar and vector-vector self-interactions.
- Performing nonrelativistic reduction of the NLDE solutions.
- Analyzing the stability and blowup criteria for the derived solitary waves.
Main Results:
- Exact analytic solitary wave solutions were found for arbitrary k.
- These solutions generalize NLSE exact solutions and reduce to them in the nonrelativistic limit.
- A 1/2m correction to the NLSE was derived for the nonrelativistic limit (|ω-m|<<2m).
Conclusions:
- The derived solitary waves are generalizations of NLSE solutions.
- The nonrelativistic reduction provides a new correction term for the NLSE.
- Solitary waves are predicted to be stable for k<2, indicating potential for robust localized structures.
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