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Radial solutions and phase separation in a system of two coupled Schrödinger equations. (English) Zbl 1161.35051
Summary: We consider the nonlinear elliptic system $\begin{cases} -\Delta u +u - u^3 -\beta v^2u = 0\quad&\text{in }\mathbb B, \\ -\Delta v +v - v^3 -\beta u^2v = 0\quad& \text{in } \mathbb B,\\ u,v > 0 \quad \text{in } \mathbb B,\quad u=v=0 \quad& \text{on } \partial \mathbb B, \end{cases}$ where $$N\leqq 3$$ and $$\mathbb B \subset \mathbb {R}^N$$ is the unit ball. We show that, for every $$\beta \leqq -1$$ and $$k \in \mathbb N$$, the above problem admits a radially symmetric solution $$(u_\beta,v_\beta)$$ such that $$u_\beta - v_\beta$$ changes sign precisely $$k$$ times in the radial variable. Furthermore, as $$\beta \to -\infty$$, after passing to a subsequence, $$u_\beta \rightarrow w ^+$$ and $$v _\beta \rightarrow w^{-}$$ uniformly in $$\mathbb B$$, where $$w = w^{+} - w^{-}$$ has precisely $$k$$ nodal domains and is a radially symmetric solution of the scalar equation $$\Delta w - w + w^3 = 0$$ in $$\mathbb B$$, $$w = 0$$ on $$\partial \mathbb B$$. Within a Hartree-Fock approximation, the result provides a theoretical indication of phase separation into many nodal domains for Bose-Einstein double condensates with strong repulsion.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 81V10 Electromagnetic interaction; quantum electrodynamics 35B40 Asymptotic behavior of solutions to PDEs 35A35 Theoretical approximation in context of PDEs
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##### References:
  Amann H. (1985) Global existence for semilinear parabolic systems. J. Reine Angew. Math. 360: 47–83 · Zbl 0564.35060 · doi:10.1515/crll.1985.360.47  Ambrosetti A., Colorado E. (2006) Bound and ground states of coupled nonlinear Schrodinger equations. C.R. Math. Acad. Sci. Paris 342: 453–458 · Zbl 1094.35112  Akhmediev N., Ankiewicz A. (1998) Partially coherent solitons on a finite background. Phys. Rev. Lett. 82: 2661–2665 · doi:10.1103/PhysRevLett.82.2661  Angenent S. (1988) The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 390: 79–96 · Zbl 0644.35050 · doi:10.1515/crll.1988.390.79  Bartsch, T., Wang, Z.-Q., Wei, J.: Bound states for a coupled Schrödinger system. preprint. · Zbl 1153.35390  Bartsch T., Willem M. (1993) Infinitely many radial solutions of a semilinear elliptic problem on R N . Arch. Ration. Mech. Anal. 124(3): 261–276 · Zbl 0790.35020 · doi:10.1007/BF00953069  Cazenave T., Lions P.-L. (1984) Solutions globales d’équations de la chaleur semi linéaires. Comm. Partial Differ. Equ. 9(10): 955–978 · Zbl 0555.35067 · doi:10.1080/03605308408820353  Chen X.Y., Poláčik P. (1996) Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball. J. Reine Angew. Math. 472: 17–51 · Zbl 0839.35059 · doi:10.1515/crll.1996.472.17  Chang S., Lin C.S., Lin T.C., Lin W. (2004) Segregated nodal domains of two-dimensional multispecies Bose–Einstein condensates. Phys. D 196(3–4): 341–361 · Zbl 1098.82602 · doi:10.1016/j.physd.2004.06.002  Christodoulides D.N., Coskun T.H., Mitchell M., Segev M. (1997) Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett. 78: 646–649 · doi:10.1103/PhysRevLett.78.646  Conti M., Merizzi L., Terracini S. (2000) Radial solutions of superlinear equations on R N . I. A global variational approach. Arch. Ration. Mech. Anal. 153(4): 291–316 · Zbl 0961.35043 · doi:10.1007/s002050050015  Conti M., Terracini S. (2000) Radial solutions of superlinear equations on R N . II. The forced case. Arch. Ration. Mech. Anal. 153(4): 317–339 · Zbl 0958.35038 · doi:10.1007/s002050050016  Conti M., Terracini S., Verzini G. (2002) Nehari’s problem and competing species systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 19(6): 871–888 · Zbl 1090.35076 · doi:10.1016/S0294-1449(02)00104-X  Conti M., Terracini S., Verzini G. (2005) Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195(2): 524–560 · Zbl 1126.35016 · doi:10.1016/j.aim.2004.08.006  Dancer E.N., Du Y. (1994) Competing species equations with diffusion, large interactions, and jumping nonlinearities. J. Differ. Equ. 114(2): 434–475 · Zbl 0815.35024 · doi:10.1006/jdeq.1994.1156  Esry B.D., Greene C.H., Burke Jr J.P., Bohn J.L. (1997) Hartree–Fock theory for double condensates. Phys. Rev. Lett. 78: 3594–3597 · doi:10.1103/PhysRevLett.78.3594  Gilbarg D., Trudinger N.S. (1983) Elliptic Partial Differential Equations of Second order, 2nd edn. Springer, Heidelberg · Zbl 0562.35001  Hall D.S., Matthews M.R., Ensher J.R., Wieman C.E., Cornell E.A. (1998) Dynamics of component separation in a binary mixture of Bose–Einstein condensates. Phys. Rev. Lett. 81: 1539–1542 · doi:10.1103/PhysRevLett.81.1539  Henry D.B. (1985) Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations. J. Differ. Equ. 59: 165–205 · Zbl 0572.58012 · doi:10.1016/0022-0396(85)90153-6  Hioe F.T. (1999) Solitary waves for N coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 82: 1152–1155 · Zbl 0930.35162 · doi:10.1103/PhysRevLett.82.1152  Hioe F.T., Salter T.S. (2002) Special set and solutions of coupled nonlinear Schrödinger equations. J. Phys. A: Math. Gen. 35: 8913–8928 · Zbl 1040.35115 · doi:10.1088/0305-4470/35/42/303  Kanna T., Lakshmanan M. (2001) Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 86: 5043–5046 · doi:10.1103/PhysRevLett.86.5043  Lin T.-C., Wei J.-C. (2005) Ground state of N coupled nonlinear Schrödinger equations in R n , $$n \leqq 3$$ . Commun. Math. Phys. 255(3): 629–653 · Zbl 1119.35087 · doi:10.1007/s00220-005-1313-x  Lin T.C., Wei J. (2005) Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(4): 403–439 · Zbl 1080.35143 · doi:10.1016/j.anihpc.2004.03.004  Lin, T.C., Wei, J.: Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations. Phys. D: Nonlinear Phenomena, to appear · Zbl 1105.35116  Maia L.A., Montefusco E., Pellacci B. (2006) Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229: 743–767 · Zbl 1104.35053 · doi:10.1016/j.jde.2006.07.002  Matano H. (1982) Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29: 401–441 · Zbl 0496.35011  Mitchell M., Chen Z., Shih M., Segev M. (1996) Self-Trapping of partially spatially incoherent light. Phys. Rev. Lett. 77: 490–493 · doi:10.1103/PhysRevLett.77.490  Mitchell M., Segev M. (1997) Self-trapping of incoherent white light. Nature 387: 880–882 · doi:10.1038/43079  Myatt C.J., Burt E.A., Ghrist R.W., Cornell E.A., Wieman C.E. (1997) Production of two overlapping Bose–Einstein condensates by sympathetic cooling. Phys. Rev. Lett. 78: 586–589 · doi:10.1103/PhysRevLett.78.586  Nehari Z. (1961) Characteristic values associated with a class of non-linear second-order differential equations. Acta Math. 105: 141–175 · Zbl 0099.29104 · doi:10.1007/BF02559588  Nickel K. (1962) Gestaltaussagen über Lösungen parabolischer Differentialgleichungen. J. Reine Angew. Math. 211: 78–94 · Zbl 0127.31801 · doi:10.1515/crll.1962.211.78  Quittner P. (2004) Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems. NoDEA Nonlinear Differ. Equ. Appl. 11: 237–258 · Zbl 1058.35120  Sattinger D.H. (1969) On the total variation of solutions of parabolic equations. Math. Ann. 183: 78–92 · Zbl 0176.40501 · doi:10.1007/BF01361263  Sirakov, B.: Least energy solitary waves for a system of nonlinear Schrödinger equations. preprint. · Zbl 1147.35098  Struwe M. (1980) Multiple solutions of anticoercive boundary value problems for a class of ordinary differential equations of second order. J. Differ. Equ. 37(2): 285–295 · Zbl 0432.34014 · doi:10.1016/0022-0396(80)90099-6  Struwe M. (1981) Infinitely many solutions of superlinear boundary value problems with rotational symmetry. Arch. Math. (Basel) 36(4): 360–369 · Zbl 0441.35010  Struwe M. (1982) Superlinear elliptic boundary value problems with rotational symmetry. Arch. Math. (Basel) 39(3): 233–240 · Zbl 0496.35034  Struwe M. (1990) Variational Methods. Springer, Berlin  Terracini S., Verzini G. (2001) Solutions of prescribed number of zeroes to a class of superlinear ODE’s systems. NoDEA Nonlinear Differ. Equ. Appl. 8: 323–341 · Zbl 0988.34013 · doi:10.1007/PL00001451  Timmermans E. (1998) Phase separation of Bose–Einstein condensates. Phys. Rev. Lett. 81: 5718–5721 · doi:10.1103/PhysRevLett.81.5718  Troy W.C. (1981) Symmetry properties in systems of semilinear elliptic equations. J. Differ. Equ. 42(3): 400–413 · Zbl 0486.35032 · doi:10.1016/0022-0396(81)90113-3  Wei, J.C., Weth, T.: Nonradial symmetric bound states for a system of coupled Schrödinger equations. preprint. · Zbl 1229.35019  Willem M. (1996) Minimax theorems. PNLDE 24, Birkhäuser, Boston · Zbl 0856.49001
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