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Symmetry breaking bifurcations in the NLS equation with an asymmetric delta potential

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Abstract

We consider the NLS equation with a linear double-well potential. Symmetry breaking, i.e. the localisation of an order parameter in one of the potential wells that can occur when the system is symmetric, has been studied extensively. However, when the wells are asymmetric, only a few analytical works have been reported. Using double Dirac delta potentials, we study rigorously the effect of such asymmetry on the bifurcation type. We show that the standard pitchfork bifurcation becomes broken, i.e. unfolded, and instead a saddle-centre type is obtained. Using a geometrical approach, we also establish the instability of the corresponding solutions along each branch in the bifurcation diagram.

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Acknowledgements

R.R gratefully acknowledges financial support from Lembaga Pengelolaan Dana Pendidikan (Indonesia Endowment Fund for Education), Grant Ref. No: S-5405/LPDP.3/2015.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, UK

    Rahmi Rusin

  2. Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia, Ged D lt. 2 FMIPA Kampus UI, Depok, 16424, Indonesia

    Rahmi Rusin

  3. School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW, 2006, Australia

    Robert Marangell

  4. Department of Mathematics, Khalifa University, PO Box 127788, Abu Dhabi, United Arab Emirates

    Hadi Susanto

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  1. Rahmi Rusin
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The authors contributed equally to the manuscript.

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Rusin, R., Marangell, R. & Susanto, H. Symmetry breaking bifurcations in the NLS equation with an asymmetric delta potential. Nonlinear Dyn 100, 3815–3824 (2020). https://doi.org/10.1007/s11071-020-05730-3

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