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Spike vector solutions for some coupled nonlinear Schrödinger equations
1. | Department of Mathematics, Central China Normal University, Wuhan 430079 |
2. | Center for Partial Differential Equations, East China Normal University, Shanghai, 200241, China |
References:
[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. |
[2] |
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. |
[3] |
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. |
[4] |
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. |
[5] |
A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equation on $\mathbb{R}^N2$, J. Funct. Anal., 254 (2008), 2816-2845.
doi: 10.1016/j.jfa.2007.11.013. |
[6] |
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. |
[7] |
T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207. |
[8] |
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. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
Z. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193. |
[23] |
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. |
[24] |
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. |
[25] |
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. |
[26] |
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. |
[27] |
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. |
[28] |
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. |
[29] |
A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.
doi: 10.1016/j.jde.2005.09.002. |
[30] |
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. |
[31] |
B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[32] |
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. |
[33] |
R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223. |
[34] |
E. Timmermans, Phase seperation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721. |
show all references
References:
[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. |
[2] |
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. |
[3] |
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. |
[4] |
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. |
[5] |
A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equation on $\mathbb{R}^N2$, J. Funct. Anal., 254 (2008), 2816-2845.
doi: 10.1016/j.jfa.2007.11.013. |
[6] |
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. |
[7] |
T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207. |
[8] |
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. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
Z. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193. |
[23] |
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. |
[24] |
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. |
[25] |
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. |
[26] |
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. |
[27] |
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. |
[28] |
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. |
[29] |
A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.
doi: 10.1016/j.jde.2005.09.002. |
[30] |
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. |
[31] |
B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[32] |
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. |
[33] |
R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223. |
[34] |
E. Timmermans, Phase seperation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721. |
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