Advanced Search
Article Contents
Article Contents

Spike vector solutions for some coupled nonlinear Schrödinger equations

Abstract Related Papers Cited by
  • We consider spike vector solutions for the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^{2}\Delta u+P(x)u=\mu u^{3}+\beta v^2u \ \hbox{in}\ \mathbb{R}^3,\\ -\varepsilon^{2}\Delta v+Q(x)v=\nu v^{3} +\beta u^2v \ \ \hbox{in}\ \mathbb{R}^3,\\ u, v >0 \,\ \hbox{in}\ \mathbb{R}^3, \end{array} \right. \end{equation*} where $\varepsilon > 0$ is a small parameter, $P(x)$ and $Q(x)$ are positive potentials, $\mu>0, \nu>0$ are positive constants and $\beta\neq 0$ is a coupling constant. We investigate the effect of potentials and the nonlinear coupling on the solution structure. For any positive integer $k\ge 2$, we construct $k$ interacting spikes concentrating near the local maximum point $x_{0}$ of $P(x)$ and $Q(x)$ when $P(x_{0})=Q(x_{0})$ in the attractive case. In contrast, for any two positive integers $k\ge 2$ and $m\ge 2$, we construct $k$ interacting spikes for $u$ near the local maximum point $x_{0}$ of $P(x)$ and $m$ interacting spikes for $v$ near the local maximum point $\bar{x}_{0}$ of $Q(x)$ respectively when $x_{0}\neq \bar{x}_{0}$, moreover, spikes of $u$ and $v$ repel each other. Meanwhile, we prove the attractive phenomenon for $\beta < 0$ and the repulsive phenomenon for $\beta > 0$.
    Mathematics Subject Classification: Primary: 35B40; Secondary: 35B20, 35J47, 35Q40.


    \begin{equation} \\ \end{equation}
  • [1]

    N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.doi: 10.1103/PhysRevLett.82.2661.


    A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.doi: 10.1016/j.crma.2006.01.024.


    A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.doi: 10.1112/jlms/jdl020.


    A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equation, Calc. Var. Partial Differ. Equ., 30 (2007), 85-112.doi: 10.1007/s00526-006-0079-0.


    A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equation on $\mathbbR^N$, J. Funct. Anal., 254 (2008), 2816-2845.doi: 10.1016/j.jfa.2007.11.013.


    T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic systemm, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.doi: 10.1007/s00526-009-0265-y.


    T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207.


    T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.doi: 10.1007/s11784-007-0033-6.


    D. Cao and S. Peng, Semi-classical bound states for Schrödinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations, 34 (2009), 1566-1591.doi: 10.1080/03605300903346721.


    S. Chang, C. S. Lin, T. C. Lin and W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Phys. D, 196 (2004), 341-361.doi: 10.1016/j.physd.2004.06.002.


    M. Conti, S. Terracini and G. Verzini, Neharis problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 871-888.doi: 10.1016/S0294-1449(02)00104-X.


    N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 1189-1208.doi: 10.1090/S0002-9947-08-04735-1.


    N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.doi: 10.1016/j.anihpc.2010.01.009.


    B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.doi: 10.1103/PhysRevLett.78.3594.


    D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. H. Poincaré Anal. NonLinéaire, 25 (2008), 149-161.doi: 10.1016/j.anihpc.2006.11.006.


    S. Gupta, Z. Hadzibabic, M. W. Zwierlein, C. A. Stan, K. Dieckmann, C. H. Schunck, E. G. M. van Kempen, B. J. Verhaar and W. Ketterle, Radio-frequency spectroscopy of ultracold fermions, Science, 300 (2003), 1723-1726.doi: 10.1126/science.1085335.


    D. S. Hall, R. Matthews, J. R. Ensher, C. E. Wieman and E. A. Cornell, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Collected Papers of Carl Wieman, (2008), 515-518.doi: 10.1142/9789812813787_0071.


    T. C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbbR^n, n\leq3$, Commun. Math. Phys., 255 (2005), 629-653.doi: 10.1007/s00220-005-1313-x.


    T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403-439.doi: 10.1016/j.anihpc.2004.03.004.


    T. C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569.doi: 10.1016/j.jde.2005.12.011.


    Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731.doi: 10.1007/s00220-008-0546-x.


    Z. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.


    L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for weakly coupled nonlinear Schrödinger systems, J. Differential Equations, 229 (2006), 743-767.doi: 10.1016/j.jde.2006.07.002.


    E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 41-71.doi: 10.4171/JEMS/103.


    C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell and C. E. Wieman, Production of two overlapping Bose-Einstein condensates by sympathetic cooling, Collected Papers of Carl Wieman, (2008), 489-492.doi: 10.1142/9789812813787_0066.


    B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.


    E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2000), 213-227.doi: 10.1112/S002461070000898X.


    S. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339.doi: 10.1007/s00205-012-0598-0.


    A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.doi: 10.1016/j.jde.2005.09.002.


    O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52.doi: 10.1016/0022-1236(90)90002-3.


    B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$, Comm. Math. Phys., 271 (2007), 199-221.doi: 10.1007/s00220-006-0179-x.


    S. Terracini and G. Verzini, Multipulse phase in $k$-mixtures of Bose-Einstein condenstates, Arch. Ration. Mech. Anal., 194 (2009), 717-741.doi: 10.1007/s00205-008-0172-y.


    R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.


    E. Timmermans, Phase seperation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.

  • 加载中

Article Metrics

HTML views() PDF downloads(111) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint