Abstract
In the exciton-polariton system, a linear dispersive photon field is coupled to a nonlinear exciton field. Short-time analysis of the lossless system shows that, when the photon field is excited, the time required for that field to exhibit nonlinear effects is longer than the time required for the nonlinear Schrödinger equation, in which the photon field itself is nonlinear. When the initial condition is scaled by \(\epsilon ^\alpha \), it is found that the relative error committed by omitting the nonlinear term in the exciton-polariton system remains within \(\epsilon \) for all times up to \(t=C\epsilon ^\beta \), where \(\beta =(1-\alpha (p-1))/(p+2)\). This is in contrast to \(\beta =1-\alpha (p-1)\) for the nonlinear Schrödinger equation. The result is proved for solutions in \(H^s(\mathbb {R}^n)\) for \(s>n/2\). Numerical computations indicate that the results are sharp and also hold in \(L^2(\mathbb {R}^n)\).
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Acknowledgements
The authors would like to thank Rodrigo Platte, Svetlana Roudenko and Kai Yang for making their code available to us for the numerical simulations. SPS acknowledges the support of NSF research Grant DMS-1411393.
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Communicated by Charles R. Doering.
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Guevara, C.D., Shipman, S.P. Short-Time Nonlinear Effects in the Exciton-Polariton System. J Nonlinear Sci 28, 579–597 (2018). https://doi.org/10.1007/s00332-017-9419-9
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DOI: https://doi.org/10.1007/s00332-017-9419-9