Converse problem for the two-component radial Gross-Pitaevskii system with a large coupling parameter

Casteras, Jean-Baptiste; Sourdis, Christos

  • Proceedings of Equadiff 14, Publisher: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing(Bratislava), page 397-406

Abstract

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We consider strongly coupled competitive elliptic systems that arise in the study of two-component Bose-Einstein condensates. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. In the case of radial symmetry, under natural non-degeneracy assumptions on a solution of the limit problem, we establish by a perturbation argument its persistence as a solution to the elliptic system.

How to cite

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Casteras, Jean-Baptiste, and Sourdis, Christos. "Converse problem for the two-component radial Gross-Pitaevskii system with a large coupling parameter." Proceedings of Equadiff 14. Bratislava: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, 2017. 397-406. <http://eudml.org/doc/294940>.

@inProceedings{Casteras2017,
abstract = {We consider strongly coupled competitive elliptic systems that arise in the study of two-component Bose-Einstein condensates. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. In the case of radial symmetry, under natural non-degeneracy assumptions on a solution of the limit problem, we establish by a perturbation argument its persistence as a solution to the elliptic system.},
author = {Casteras, Jean-Baptiste, Sourdis, Christos},
booktitle = {Proceedings of Equadiff 14},
keywords = {Singular perturbation, competitive elliptic system, segregation},
location = {Bratislava},
pages = {397-406},
publisher = {Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing},
title = {Converse problem for the two-component radial Gross-Pitaevskii system with a large coupling parameter},
url = {http://eudml.org/doc/294940},
year = {2017},
}

TY - CLSWK
AU - Casteras, Jean-Baptiste
AU - Sourdis, Christos
TI - Converse problem for the two-component radial Gross-Pitaevskii system with a large coupling parameter
T2 - Proceedings of Equadiff 14
PY - 2017
CY - Bratislava
PB - Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing
SP - 397
EP - 406
AB - We consider strongly coupled competitive elliptic systems that arise in the study of two-component Bose-Einstein condensates. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. In the case of radial symmetry, under natural non-degeneracy assumptions on a solution of the limit problem, we establish by a perturbation argument its persistence as a solution to the elliptic system.
KW - Singular perturbation, competitive elliptic system, segregation
UR - http://eudml.org/doc/294940
ER -

References

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  1. Aftalion, A., Sourdis, C., Interface layer of a two-component Bose-Einstein condensate, , Commun. Contemp. Math. 19 (2017), 1650052. MR3670791
  2. Aftalion, A., Pacella, F., Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball, , J. Differential Equations 195 (2003), pp. 380-397. MR2016817
  3. Ao, W., Wei, J., Yao, W., Uniqueness and nondegeneracy of sign-changing radial solutions to an almost critical elliptic problem, , Advances in Differential Equations 21 (2016), pp. 1049–1084. MR3556760
  4. Berestycki, H., Lin, T-C., Wei, J., Zhao, C., On phase-separation models: asymptotics and qualitative properties, , Arch. Ration. Mech. Anal. 208 (2013), pp. 163–200. MR3021546
  5. Berestycki, H., Terracini, S., Wang, K., Wei, J., On entire solutions of an elliptic system modeling phase separations, , Adv. Math. 243 (2013), pp. 102–126. MR3062741
  6. Conti, M., Terracini, S., Verzini, G., Asymptotic estimates for the spatial segregation of competitive systems, , Adv. Math. 195 (2005), pp. 524-560. MR2146353
  7. Dancer, E. N., Du, Y., Competing species equations with diffusion, large interactions, andjumping nonlinearities, , J. Differential Equations 114 (1994), pp. 434–475. MR1303035
  8. Dancer, E. N., Wang, K., Zhang, Z., Uniform Hölder estimate for singularly perturbed parabolic systems of Bose–Einstein condensates and competing species, , J. Differential Equations 251 (2011), pp. 2737–2769. MR2831712
  9. Dancer, E. N., Wang, K., Zhang, Z., The limit equation for the Gross-Pitaevskii equations and S. Terracini’s conjecture, , J. Functional Analysis 262 (2012), pp. 1087–1131. MR2863857
  10. Dancer, E. N., On the converse problem for the Gross-Pitaevskii equations with a large parameter, , Discr. Cont. Dyn. Syst. 34 (2014), pp. 2481–2493. MR3177644
  11. Dancer, E. N., Slides, , https://math.umons.ac.be/anum/pde2015/documents/Dancer.pdf. 
  12. Felmer, P., Martinez, S., Tanaka, K., Uniqueness of radially symmetric positive solutions for − Δ u + u = u p in an annulus, , J. Differential Equations 245 (2008), pp. 1198–1209. MR2436828
  13. Noris, B., Tavares, H., Terracini, S., Verzini, G., Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, , Comm. Pure Appl. Math. 63 (2010), pp. 267–302. MR2599456
  14. Pacella, F., Uniqueness of positive solutions of semilinear elliptic equations and related eigenvalue problems, , Milan Journal of Mathematics 73 (2005), pp. 221–236. MR2175043
  15. Santos, E. Moreira dos, Pacella, F., Morse index of radial nodal solutions of Hénon type equations in dimension two, , Communications in Contemporary Mathematics 19 (2017), 1650042. MR3631930
  16. Shioji, N., Watanabe, K., A generalized Pohožaev identity and uniqueness of positive radial solutions of Δ u + g ( r ) u + h ( r ) u p = 0 , , J. Differential Equations 255 (2013), pp. 4448–4475. MR3105928
  17. Shioji, N., Watanabe, K., Uniqueness and nondegeneracy of positive radial solutions of div ( ρ ∇ u ) + ρ ( − g u + h u p ) = 0 , , Calc. Var. Partial Differential Equations 55 (2016), 42pp. MR3470747
  18. Soave, N., Zilio, A., Uniform bounds for strongly competing systems: The optimal Lipschitz case, , Arch. Ration. Mech. Anal. 218 (2015), pp. 647–697. MR3375537
  19. Soave, N., Zilio, A., Multidimensional entire solutions for an elliptic system modelling phase separation, , Analysis and PDE 9 (2016), pp. 1019-1041. MR3531365
  20. Soave, N., Zilio, A., On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects, , Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), pp. 625–654. MR3633738
  21. Tanaka, S., Uniqueness of sign-changing radial solutions for Δ u − u + | u | p − 1 u = 0 in some ball and annulus, , J. Math. Anal. Appl. 439 (2016), pp. 154–170. MR3474355
  22. Tavares, H., Terracini, S., Regularity of the nodal set of segregated critical configurations under a weak reflection law, , Calc. Var. 45 (2012), pp. 273–317. MR2984134
  23. Zhang, S., Liu, Z., Singularities of the nodal set of segregated configurations, , Calc. Var. 54 (2015), pp. 2017–2037. MR3396442
  24. Wang, K., Uniform Lipschitz regularity of flat segregated interfaces in a singularly perturbed problem, , Calc. Var. (2017) 56:135. MR3690006
  25. Wei, J., Weth, T., Asymptotic behaviour of solutions of planar elliptic systems with strong competition, , Nonlinearity 21 (2008), pp. 305–317. MR2384550
  26. Wei, J., Weth, T., Radial solutions and phase separation in a system of two coupled Schrödinger equations, , Arch. Ration. Mech. Anal. 190 (2008), pp. 83-106. MR2434901

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