# American Institute of Mathematical Sciences

April  2016, 36(4): 2205-2227. doi: 10.3934/dcds.2016.36.2205

## 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

Received  February 2014 Revised  July 2015 Published  September 2015

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 $\beta > 0$.
Citation: Shuangjie Peng, Huirong Pi. Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2205-2227. doi: 10.3934/dcds.2016.36.2205
##### 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.
 [1] Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259 [2] Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201 [3] Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739 [4] Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383 [5] Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003 [6] Chuangye Liu, Rushun Tian. Normalized solutions for 3-coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5115-5130. doi: 10.3934/cpaa.2020229 [7] Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911 [8] Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118 [9] Thierry Cazenave, Zheng Han. Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4801-4819. doi: 10.3934/dcds.2020202 [10] Jing Yang. Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1785-1805. doi: 10.3934/cpaa.2017087 [11] Jing Cui, Shu-Ming Sun. Nonlinear Schrödinger equations on a finite interval with point dissipation. Mathematical Control and Related Fields, 2019, 9 (2) : 351-384. doi: 10.3934/mcrf.2019017 [12] Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921 [13] Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure and Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883 [14] Lie Zheng. Asymptotic behavior of solutions to the nonlinear breakage equations. Communications on Pure and Applied Analysis, 2005, 4 (2) : 463-473. doi: 10.3934/cpaa.2005.4.463 [15] Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138 [16] Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265 [17] Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789 [18] M. D. Todorov, C. I. Christov. Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations. Conference Publications, 2007, 2007 (Special) : 982-992. doi: 10.3934/proc.2007.2007.982 [19] Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93 [20] Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure and Applied Analysis, 2021, 20 (2) : 867-884. doi: 10.3934/cpaa.2020294

2021 Impact Factor: 1.588