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Remarks on the critical coupling strength for the Cucker-Smale model with unit speed
Synchronization of positive solutions for coupled Schrödinger equations
1. | School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Luo-Yu Road 152, Wuhan 430079, China |
2. | Center for Applied Mathematics, Tianjin University, Tianjin 300072, China |
3. | Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA |
$\left\{ {\begin{array}{*{20}{c}} {\Delta u - u + ({\mu _1}|u{|^p} + \beta |v{|^p})|u{|^{p - 2}}u = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \\ {\Delta v - v + ({\mu _2}|v{|^p} + \beta |u{|^p})|v{|^{p - 2}}v = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \end{array}} \right.$ |
$ 2< p<\frac{n}{n-2}, $ |
$ n\ge 3$ |
$ 2< p<+∞ $ |
$ n = 1, 2, $ |
$μ_1, μ_2, β>0 $ |
$ n = 1 $ |
$ p = 2 $ |
References:
[1] |
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. |
[2] |
T. Bartsch and Z.-Q. Wang,
Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207.
|
[3] |
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. |
[4] |
T. Bartsch, N. Dancer and Z.-Q. Wang,
A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[5] |
D. J. Benney and A. C. Newell,
The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.
doi: 10.1002/sapm1967461133. |
[6] |
S. Correia,
Characterization of ground-states for a system of M coupled semilinear Schrödinger equations and applications, J.Differential Equations, 260 (2016), 3302-3326.
doi: 10.1016/j.jde.2015.10.032. |
[7] |
D. G. de Figueiredo and O. Lopes,
Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 149-161.
doi: 10.1016/j.anihpc.2006.11.006. |
[8] |
A. Hasegawa and F. Tappert,
Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅰ. Anormalous dispersion, Appl. Phys. Lett., 23 (1973), 142-144.
|
[9] |
A. Hasegawa and F. Tappert,
Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅱ. Normal dispersion, Appl. Phys. Lett., 23 (1973), 171-173.
doi: 10.1063/1.1654847. |
[10] |
T. Lin and J. Wei,
Ground state of N coupled nonlinear Schrödinger equations in $ \mathbb{R}^n, n\le 3 $, Communications in Mathematical Physics, 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[11] |
H. Liu, Z. Liu and J. Chang,
Existence and uniquiness of positive solutions of nonlinear Schrödinger systems, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 365-390.
doi: 10.1017/S0308210513000711. |
[12] |
R. Mandel,
Minimal energy solutions for repulsive nonlinear Schrödinger systems, J. Differential Equations, 257 (2014), 450-468.
doi: 10.1016/j.jde.2014.04.006. |
[13] |
M. Mitchell and M. Segev,
Self-trapping of inconherent white light, Nature, 387 (1997), 880-882.
|
[14] |
G. J. Roskes,
Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238.
doi: 10.1002/sapm1976553231. |
[15] |
C. Rüegg et al,
Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.
|
[16] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations in $ R^n $, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[17] |
W. C. Troy,
Symmetry properties in systems of semilinear elliptic equations, Journal of Differential Equations, 42 (1981), 400-413.
doi: 10.1016/0022-0396(81)90113-3. |
[18] |
Z.-Q. Wang and M. Willem,
Partial symmetry of vector solutions for elliptic systems, Journal d'Analyse Mathématique, 122 (2014), 69-85.
doi: 10.1007/s11854-014-0003-z. |
[19] |
J. Wei and W. Yao,
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
[20] |
J. Yang,
Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152.
doi: 10.1111/1467-9590.00073. |
[21] |
V. E. Zakharov,
Stability of periodic waves of finite amplitude on the surface of a deep fluid, Sov. Phys. J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.
doi: 10.1007/BF00913182. |
[22] |
V. E. Zakharov,
Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914.
|
show all references
References:
[1] |
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. |
[2] |
T. Bartsch and Z.-Q. Wang,
Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207.
|
[3] |
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. |
[4] |
T. Bartsch, N. Dancer and Z.-Q. Wang,
A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[5] |
D. J. Benney and A. C. Newell,
The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.
doi: 10.1002/sapm1967461133. |
[6] |
S. Correia,
Characterization of ground-states for a system of M coupled semilinear Schrödinger equations and applications, J.Differential Equations, 260 (2016), 3302-3326.
doi: 10.1016/j.jde.2015.10.032. |
[7] |
D. G. de Figueiredo and O. Lopes,
Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 149-161.
doi: 10.1016/j.anihpc.2006.11.006. |
[8] |
A. Hasegawa and F. Tappert,
Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅰ. Anormalous dispersion, Appl. Phys. Lett., 23 (1973), 142-144.
|
[9] |
A. Hasegawa and F. Tappert,
Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅱ. Normal dispersion, Appl. Phys. Lett., 23 (1973), 171-173.
doi: 10.1063/1.1654847. |
[10] |
T. Lin and J. Wei,
Ground state of N coupled nonlinear Schrödinger equations in $ \mathbb{R}^n, n\le 3 $, Communications in Mathematical Physics, 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[11] |
H. Liu, Z. Liu and J. Chang,
Existence and uniquiness of positive solutions of nonlinear Schrödinger systems, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 365-390.
doi: 10.1017/S0308210513000711. |
[12] |
R. Mandel,
Minimal energy solutions for repulsive nonlinear Schrödinger systems, J. Differential Equations, 257 (2014), 450-468.
doi: 10.1016/j.jde.2014.04.006. |
[13] |
M. Mitchell and M. Segev,
Self-trapping of inconherent white light, Nature, 387 (1997), 880-882.
|
[14] |
G. J. Roskes,
Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238.
doi: 10.1002/sapm1976553231. |
[15] |
C. Rüegg et al,
Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.
|
[16] |
B. Sirakov,
Least energy solitary waves for a system of nonlinear Schrödinger equations in $ R^n $, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[17] |
W. C. Troy,
Symmetry properties in systems of semilinear elliptic equations, Journal of Differential Equations, 42 (1981), 400-413.
doi: 10.1016/0022-0396(81)90113-3. |
[18] |
Z.-Q. Wang and M. Willem,
Partial symmetry of vector solutions for elliptic systems, Journal d'Analyse Mathématique, 122 (2014), 69-85.
doi: 10.1007/s11854-014-0003-z. |
[19] |
J. Wei and W. Yao,
Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.
doi: 10.3934/cpaa.2012.11.1003. |
[20] |
J. Yang,
Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152.
doi: 10.1111/1467-9590.00073. |
[21] |
V. E. Zakharov,
Stability of periodic waves of finite amplitude on the surface of a deep fluid, Sov. Phys. J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.
doi: 10.1007/BF00913182. |
[22] |
V. E. Zakharov,
Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914.
|


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