June  2018, 38(6): 2795-2808. doi: 10.3934/dcds.2018118

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

* Corresponding author: Zhi-Qiang Wang

Received  May 2017 Published  April 2018

In this paper, we analyze synchronized positive solutions for a coupled nonlinear Schrödinger equation
$\left\{ {\begin{array}{*{20}{c}} {\Delta u - u + ({\mu _1}|u{|^p} + \beta |v{|^p})|u{|^{p - 2}}u = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \\ {\Delta v - v + ({\mu _2}|v{|^p} + \beta |u{|^p})|v{|^{p - 2}}v = 0,}&{{\text{i}}n\;{\mathbb{R}^n},} \end{array}} \right.$
where
$ 2< p<\frac{n}{n-2}, $
if
$ n\ge 3$
and
$ 2< p<+∞ $
, if
$ n = 1, 2, $
and
$μ_1, μ_2, β>0 $
are positive constants. Our goal is two fold. On one hand we study the question under what conditions the ground states are nontrivial synchronized positive solutions, giving precise conditions in terms of the size of the coupling constant. On the other hand, we examine the questions on whether all positive solutions are synchronized solutions. We have a complete answer for the case
$ n = 1 $
by proving that positivity implies synchronization. The latter result enables us to obtain the exact number of positive solutions even though no uniqueness result holds in the case, and this is quite different from the case
$ p = 2 $
for which uniqueness of positive solutions was known ([19]).
Citation: Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118
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.  Google Scholar

[2]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207.   Google Scholar

[3]

T. BartschZ.-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.  Google Scholar

[4]

T. BartschN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. LiuZ. 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.  Google Scholar

[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.  Google Scholar

[13]

M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-882.   Google Scholar

[14]

G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238.  doi: 10.1002/sapm1976553231.  Google Scholar

[15]

C. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Yang, Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152.  doi: 10.1111/1467-9590.00073.  Google Scholar

[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.  Google Scholar

[22]

V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914.   Google Scholar

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.  Google Scholar

[2]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207.   Google Scholar

[3]

T. BartschZ.-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.  Google Scholar

[4]

T. BartschN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. LiuZ. 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.  Google Scholar

[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.  Google Scholar

[13]

M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-882.   Google Scholar

[14]

G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238.  doi: 10.1002/sapm1976553231.  Google Scholar

[15]

C. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Yang, Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152.  doi: 10.1111/1467-9590.00073.  Google Scholar

[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.  Google Scholar

[22]

V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914.   Google Scholar

Figure 1.  The graphs of function $\beta = \frac{r^4-2}{r^3-r}$ (left) and $\beta = \frac{r^4-1}{r^3-r}$(right)
Figure 2.  The graphs of function $f$ in various cases
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