Spikes in two coupled nonlinear Schrödinger equations

Lin, Tai-Chia; Wei, Juncheng

Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 4, pp. 403-439

[1] Bates P., Fusco G., Equilibria with many nuclei for the Cahn-Hilliard equation, J. Differential Equations 160 (2000) 283-356. | Zbl | MR

[2] Bates P., Dancer E.N., Shi J., Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Differential Equations 4 (1999) 1-69. | Zbl | MR

[3] Conti M., Terracini S., Verzini G., Nehari's problem and competing species system, Ann. Inst. H. Poincaré 19 (6) (2002) 871-888. | Zbl | MR | Numdam

[4] Del Pino M., Felmer P., Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (3) (1999) 883-898. | Zbl | MR

[5] Del Pino M., Felmer P., Wei J., On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal. 31 (1999) 63-79. | Zbl | MR

[6] Del Pino M., Felmer P., Wei J., On the role of distance function in some singularly perturbed problems, Comm. Partial Differential Equations 25 (2000) 155-177. | Zbl | MR

[7] Del Pino M., Felmer P., Wei J., Multiple peak solutions for some singular perturbation problems, Cal. Var. Partial Differential Equations 10 (2000) 119-134. | Zbl | MR

[8] Donley E.A., Claussen N.R., Cornish S.L., Roberts J.L., Cornell E.A., Wieman C.E., Dynamics of collapsing and exploding Bose-Einstein condensates, Nature 19 (412) (2001) 295-299.

[9] Dancer E.N., Wei J., On the location of spike s of solutions with two sharp layers for a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 157 (1999) 82-101. | Zbl | MR

[10] Dancer E.N., Yan S., Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math. 189 (1999) 241-262. | Zbl | MR

[11] Estaban M.J., Lions P.L., Existence and non-existence results for semilinear problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982) 1-14. | Zbl | MR

[12] Esry B.D., Greene C.H., Burke J.P., Bohn J.L., Hartree-Fock theory for double condensates, Phys. Rev. Lett. 78 (1997) 3594-3597.

[13] Gidas B., Ni W.-M., Nirenberg L., Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, in: Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Stud., vol. 7A, Academic Press, New York, 1981, pp. 369-402. | Zbl | MR

[14] Gui C., Wei J., Multiple interior spike solutions for some singular perturbed Neumann problems, J. Differential Equations 158 (1999) 1-27. | Zbl | MR

[15] Gui C., Wei J., On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (2000) 522-538. | Zbl | MR

[16] Gui C., Wei J., Winter M., Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 249-289. | Zbl | MR | Numdam

[17] Grossi M., Pistoia A., Wei J., Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Cal. Var. Partial Differential Equations 11 (2000) 143-175. | Zbl | MR

[18] Gupta S., Hadzibabic Z., Zwierlein M.W., Stan C.A., Dieckmann K., Schunck C.H., Van Kempen E.G.M., Verhaar B.J., Ketterle W., Radio-frequency spectroscopy of ultracold fermions, Science 300 (2003) 1723-1726.

[19] Hall D.S., Matthews M.R., Ensher J.R., Wieman C.E., Cornell E.A., Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett. 81 (1998) 1539-1542.

[20] Kwong M.K., Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal. 105 (1989) 243-266. | Zbl | MR

[21] Li Y.-Y., On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations 23 (1998) 487-545. | Zbl | MR

[22] Li Y.-Y., Nirenberg L., The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math. 51 (1998) 1445-1490. | Zbl | MR

[23] Lieband E., Loss M., Analysis, American Mathematical Society, 1996. | Zbl

[24] Myatt C.J., Burt E.A., Ghrist R.W., Cornell E.A., Wieman C.E., Production of two overlapping Bose-Einstein condensates by sympathetic cooling, Phys. Rev. Lett. 78 (1997) 586-589.

[25] Ni W.-M., Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998) 9-18. | Zbl | MR

[26] Ni W.-M., Takagi I., On the shape of least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 41 (1991) 819-851. | Zbl | MR

[27] Ni W.-M., Takagi I., Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993) 247-281. | Zbl | MR

[28] Ni W.-M., Wei J., On the location and profile of spike-Layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995) 731-768. | Zbl | MR

[29] Timmermans E., Phase separation of Bose-Einstein condensates, Phys. Rev. Lett. 81 (1998) 5718-5721.

[30] Troy W., Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations 42 (3) (1981) 400-413. | Zbl | MR

[31] Wei J., On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 129 (1996) 315-333. | Zbl | MR

[32] Wei J., On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku Math. J. 50 (1998) 159-178. | Zbl | MR

[33] Wei J., On the effect of the domain geometry in a singularly perturbed Dirichlet problem, Differential Integral Equations 13 (2000) 15-45. | Zbl | MR

[34] Wei J., On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Differential Equations 134 (1997) 104-133. | Zbl | MR

[35] Wei J., Winter M., Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 459-492. | Zbl | MR | Numdam

[36] Wei J., Winter M., Multiple boundary spike solutions for a wide class of singular perturbation problems, J. London Math. Soc. 59 (1999) 585-606. | Zbl | MR