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January  2017, 16(1): 115-130. doi: 10.3934/cpaa.2017005

## A complete classification of ground-states for a coupled nonlinear Schrödinger system

 1 School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, P.R. China 2 Center for Applied Mathematics, Tianjin University, Tianjin 300072, P.R.China 3 Department of Mathematics and Statistics, Utah State University, Logan UT 84322, USA

Received  June 2016 Revised  August 2016 Published  November 2016

In this paper, we establish the existence of nontrivial ground-state solutions for a coupled nonlinear Schrödinger system
 $-\Delta u_j+ u_j=\sum\limits_{i=1}^mb_{ij}u_i^2u_j, \quad\text{in}\ \mathbb{R}^n,\\ u_j(x)\to 0\ \text{as}\ |x|\ \to \infty, \quad j=1,2,\cdots, m,$
where $n=1, 2, 3, m\geq 2$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}.$ By nontrivial we mean a solution that has all components non-zero. Due to possible systems collapsing it is important to classify ground state solutions. For $m=3$, we get a complete picture that describes whether nontrivial ground-state solutions exist or not for all possible cases according to some algebraic conditions of the matrix $B = (b_{ij})$. In particular, there is a nontrivial ground-state solution provided that all coupling constants $b_{ij}, i\neq j$ are sufficiently large as opposed to cases in which any ground-state solution has at least a zero component when $b_{ij}, i\neq j$ are all sufficiently small. Moreover, we prove that any ground-state solution is synchronized when matrix $B=(b_{ij})$ is positive semi-definite.
Citation: Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005
##### References:
 [1] J. Albert and S. Bhattarai, Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system, Advances in Differential Equations, 18 (2013), 1129-1164.  Google Scholar [2] A. Ambrosetti, 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 [3] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207.  Google Scholar [4] D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.  Google Scholar [5] 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 [6] A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅱ. Normal dispersion, Appl. Phys. Lett., 23 (1973), 171-172. Google Scholar [7] T. C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^N$, n ≤ 3, Communications in Mathematical Physics, 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.  Google Scholar [8] T. C. Lin and J. Wei, Ground State of N coupled nonlinear Schrödinger equations in $\mathbb{R}^N$, n ≤ 3, Communications in Mathematical Physics, 277 (2008), 573-576. doi: 10.1007/s00220-007-0365-5.  Google Scholar [9] 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.  Google Scholar [10] Z. Liu, Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.  Google Scholar [11] M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-882. Google Scholar [12] N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Diff. Eqns., 16 (2011), 977-1000.  Google Scholar [13] G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238. Google Scholar [14] CH. Rüegg, N.Cavadini, A. Furrer, et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. Google Scholar [15] J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.  Google Scholar [16] Z.-Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, Journal d'Analyse Mathematique, 122 (2014), 69-85. doi: 10.1007/s11854-014-0003-z.  Google Scholar [17] J. Yang, Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152. doi: 10.1111/1467-9590.00073.  Google Scholar [18] V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Sov. Phys. J. Appl. Mech. Tech. Phys., 4 (1968), 190-194. Google Scholar [19] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914. Google Scholar

show all references

##### References:
 [1] J. Albert and S. Bhattarai, Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system, Advances in Differential Equations, 18 (2013), 1129-1164.  Google Scholar [2] A. Ambrosetti, 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 [3] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J.Partial Differential Equations, 19 (2006), 200-207.  Google Scholar [4] D. J. Benney and A. C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), 133-139.  Google Scholar [5] 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 [6] A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers Ⅱ. Normal dispersion, Appl. Phys. Lett., 23 (1973), 171-172. Google Scholar [7] T. C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^N$, n ≤ 3, Communications in Mathematical Physics, 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.  Google Scholar [8] T. C. Lin and J. Wei, Ground State of N coupled nonlinear Schrödinger equations in $\mathbb{R}^N$, n ≤ 3, Communications in Mathematical Physics, 277 (2008), 573-576. doi: 10.1007/s00220-007-0365-5.  Google Scholar [9] 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.  Google Scholar [10] Z. Liu, Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.  Google Scholar [11] M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-882. Google Scholar [12] N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Diff. Eqns., 16 (2011), 977-1000.  Google Scholar [13] G. J. Roskes, Some nonlinear multiphase interactions, Stud. Appl. Math., 55 (1976), 231-238. Google Scholar [14] CH. Rüegg, N.Cavadini, A. Furrer, et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65. Google Scholar [15] J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.  Google Scholar [16] Z.-Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, Journal d'Analyse Mathematique, 122 (2014), 69-85. doi: 10.1007/s11854-014-0003-z.  Google Scholar [17] J. Yang, Multiple permanent-wave trains in nonlinear systems, Stud. Appl. Math., 100 (1998), 127-152. doi: 10.1111/1467-9590.00073.  Google Scholar [18] V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Sov. Phys. J. Appl. Mech. Tech. Phys., 4 (1968), 190-194. Google Scholar [19] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp., 35 (1972), 908-914. Google Scholar
The number of non-zero components of ground-state solutions
 case condition 1 condition 2 type 1 $det(B) > 0$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$ $p= 3$ 2 $det(B) > 0$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$ $p=1$ 3 $det(B) < 0$ $p=1, 2$ 4 $rank(B)=1$ $p=1, 2, 3$ 5 $rank(B)=2$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$ $p=2, 3$ 6 $rank(B)=2$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$ $p=1, 2$ 7 $rank(B)=2$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$ $p=1, 2, 3$ 8 $rank(B)=2$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$ $p=1$
 case condition 1 condition 2 type 1 $det(B) > 0$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$ $p= 3$ 2 $det(B) > 0$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$ $p=1$ 3 $det(B) < 0$ $p=1, 2$ 4 $rank(B)=1$ $p=1, 2, 3$ 5 $rank(B)=2$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$ $p=2, 3$ 6 $rank(B)=2$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$ $p=1, 2$ 7 $rank(B)=2$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$ $p=1, 2, 3$ 8 $rank(B)=2$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$ $p=1$
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