# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] 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 [2] 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 [3] Akhmediev N., Ankiewicz A. (1998) Partially coherent solitons on a finite background. Phys. Rev. Lett. 82: 2661–2665 · doi:10.1103/PhysRevLett.82.2661 [4] 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 [5] Bartsch, T., Wang, Z.-Q., Wei, J.: Bound states for a coupled Schrödinger system. preprint. · Zbl 1153.35390 [6] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] Gilbarg D., Trudinger N.S. (1983) Elliptic Partial Differential Equations of Second order, 2nd edn. Springer, Heidelberg · Zbl 0562.35001 [18] 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 [19] 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 [20] 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 [21] 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 [22] 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 [23] 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 [24] 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 [25] 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 [26] 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 [27] 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 [28] 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 [29] Mitchell M., Segev M. (1997) Self-trapping of incoherent white light. Nature 387: 880–882 · doi:10.1038/43079 [30] 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 [31] 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 [32] 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 [33] 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 [34] 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 [35] Sirakov, B.: Least energy solitary waves for a system of nonlinear Schrödinger equations. preprint. · Zbl 1147.35098 [36] 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 [37] Struwe M. (1981) Infinitely many solutions of superlinear boundary value problems with rotational symmetry. Arch. Math. (Basel) 36(4): 360–369 · Zbl 0441.35010 [38] Struwe M. (1982) Superlinear elliptic boundary value problems with rotational symmetry. Arch. Math. (Basel) 39(3): 233–240 · Zbl 0496.35034 [39] Struwe M. (1990) Variational Methods. Springer, Berlin [40] 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 [41] Timmermans E. (1998) Phase separation of Bose–Einstein condensates. Phys. Rev. Lett. 81: 5718–5721 · doi:10.1103/PhysRevLett.81.5718 [42] 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 [43] Wei, J.C., Weth, T.: Nonradial symmetric bound states for a system of coupled Schrödinger equations. preprint. · Zbl 1229.35019 [44] Willem M. (1996) Minimax theorems. PNLDE 24, Birkhäuser, Boston · Zbl 0856.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.