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On fractional nonlinear Schrödinger equation with combined power-type nonlinearities
Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations
1. | Department of Mathematics, Inha University, Incheon, 22212, Republic of Korea |
2. | Department of Mathematics, Chung-Ang University, Seoul, 06974, Republic of Korea |
We propose the Chern-Simons gauged sigma model from $\mathbb{R}^{1+2}$ into the hyperbolic plane $\mathbb{H}^2$. We seek a static configuration of this model and derive self-dual equations. We also establish some existence results for solutions of the self-dual equations under appropriate boundary conditions near $\infty$
References:
[1] |
J. P. Antoine and B. Piette,
Solutions of Euclidean σ models on noncompact Grassmannian manifolds, J. Math. Phys., 29 (1988), 1687-1697.
doi: 10.1063/1.527917. |
[2] |
K. Arthur, D. Tchrakian and Y. Yang,
Topological and nontopological self-dual Chern-Simons solitions in a gauged O(3) σ model, Phys. Rev. D, 54 (1996), 5245-5248.
doi: 10.1103/PhysRevD.54.5245. |
[3] |
D. Bartolucci, C.-C. Chen, C. S. Lin and G. Tarantello,
Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations, 29 (2004), 1241-1265.
doi: 10.1081/PDE-200033739. |
[4] |
D. Bartolucci, Y. Lee, C. S. Lin and M. Onodera,
Asymptotic analysis of solutions to a gauged O(3) sigma model, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 32 (2015), 651-685.
doi: 10.1016/j.anihpc.2014.03.001. |
[5] |
D. Bartolucci and G. Tarantello,
Liouville Type equations with singular data and their applications to periodic multivortices for the electroweak Theory, Comm. Math. Phys., 229 (2000), 3-47.
doi: 10.1007/s002200200664. |
[6] |
A. A. Belavin and A. M. Polyakov,
Metastable states of two dimensional isotropic ferromagnets, JETP Lett., 22 (1975), 245-247.
|
[7] |
H. Brezis and J-M. Coron,
Large solutions for harmonic maps in two dimensions, Comm. Math. Phys., 92 (1983), 203-215.
doi: 10.1007/BF01210846. |
[8] |
H. Brezis and F. Merle,
Uniform estimates and blowup behavior for solutions of $-\Delta u = V(x)e^u$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.
doi: 10.1080/03605309108820797. |
[9] |
H. Chan, C.-C. Fu and C.-S. Lin,
Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equation, Comm. Math. Phys., 231 (2002), 189-221.
doi: 10.1007/s00220-002-0691-6. |
[10] |
W. Chen and C. Li,
Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[11] |
W. Chen and C. Li,
Qualitative properties of solutions to some nonlinear elliptic equations in $\mathbb{R}^2$, Duke Math. J., 71 (1993), 427-439.
doi: 10.1215/S0012-7094-93-07117-7. |
[12] |
X. Chen, S. Hastings, J. B. McLeod and Y. Yang,
A nonlinear problem elliptic equation arising from gauge field theory and cosmology, Proc. Roy. Soc. London Ser. A, 446 (1994), 453-478.
doi: 10.1098/rspa.1994.0115. |
[13] |
K. Choe, J. Han, C.-S. Lin and T.-C. Lin,
Uniqueness and solution structure of nonlinear equations arising from the Chern-Simons gauged O(3) sigma models, J. Differential Equations, 255 (2013), 2136-2166.
doi: 10.1016/j.jde.2013.06.010. |
[14] |
K. Choe, J. Han and C.-S. Lin,
Bubbling solutions for the Chern-Simons gauged O(3) sigma modelin $\mathbb{R}^2$, Discrete Contin. Dyn. Syst., 34 (2014), 2703-2728.
doi: 10.3934/dcds.2014.34.2703. |
[15] |
K. Choe, J. Han, Y. Lee and C.-S. Lin,
Bubbling solutions for the Chern-Simons gauged O(3) sigma model on a torus, Calc. Var. Partial Differential Equations, 54 (2015), 1275-1329.
doi: 10.1007/s00526-015-0825-2. |
[16] |
K. Choe, N. Kim and C.-S. Lin,
Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 837-852.
doi: 10.1016/j.anihpc.2011.06.003. |
[17] |
P. K. Ghosh and S. K. Ghosh,
Topological and nontopological solitons in a gauged O(3) sigma model with Chern-Simons term, Phys. Lett. B, 366 (1996), 199-204.
doi: 10.1016/0370-2693(95)01365-2. |
[18] |
A. C. Huang, Harmonic maps of punctured surfaces to hyperbolic plane, preprint, arXiv: 1605.07715v1. |
[19] |
H. Huh and G. Jin, Local and global solutions of Chern-Simons gauged O(3) sigma equations in one space dimension, J. Math. Phys., 57 (2016), 081511, 11pp.
doi: 10.1063/1.4960744. |
[20] |
E. Hulett,
Harmonic superconformal maps of surfaces in $\Bbb{H}^n$, J. Geo. Phys., 42 (2002), 139-165.
doi: 10.1016/S0393-0440(01)00082-1. |
[21] |
A. Jaffe and C. Taubes, Vortices and Monopoles, Birkhäuser Boston, 1980. |
[22] |
S.-S. Kim and P. Oh,
On the gauged noncompact spin system, International Journal of Modern Physics A, 13 (1998), 5503-5517.
doi: 10.1142/S0217751X9800250X. |
[23] |
K. Kimm, K. Lee and T. Lee,
Anyonic Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Rev. D, 53 (1996), 4436-4440.
|
[24] |
J. Krieger,
Global regularity of wave maps from $\mathbb{R}^{2+1}$ to $\mathbb{H}^2$. Small energy, Comm. Math. Phys., 250 (2004), 507-580.
doi: 10.1007/s00220-004-1088-5. |
[25] |
A. Kundu,
On σ-models with noncompact groups, Lett. Math. Phys., 6 (1982), 479-485.
doi: 10.1007/BF00405869. |
[26] |
A. Kundu,
Gauge equivalence of sigma models with non-compact Grassmannian manifolds, J. Phys. A: Math. Gen., 19 (1986), 1303-1314.
doi: 10.1088/0305-4470/19/8/012. |
[27] |
P. O. Mazur,
A relationship between the electrovacuum Ernst equations and nonlinear σ-model, Acta Phys. Polon. B, 14 (1983), 219-234.
|
[28] |
P. O. Mazur,
A global identity for nonlinear σ-models, Phys. Lett. A, 100 (1984), 341-344.
doi: 10.1016/0375-9601(84)91084-3. |
[29] |
R.-C. McOwen,
Conformal metrics in $\mathbb{R}^2$ with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J., 34 (1985), 97-104.
doi: 10.1512/iumj.1985.34.34005. |
[30] |
L. Nirenberg, Topics in Nonlinear Functional Analysis, American Mathematical Society, 2001.
doi: 10.1090/cln/006. |
[31] |
P. Oh,
Bogomol'nyi solitons and Hermitian symmetric spaces, Rep. Math. Phys., 43 (1999), 271-281.
doi: 10.1016/S0034-4877(99)80035-4. |
[32] |
I. Rodnianski and J. Sterbenz,
On the formation of singularities in the critical O(3) σ-model, Ann. of Math., 172 (2010), 187-242.
doi: 10.4007/annals.2010.172.187. |
[33] |
B. J. Schroers,
Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Lett. B, 356 (1995), 291-296.
doi: 10.1016/0370-2693(95)00833-7. |
[34] |
J. Shatah and A. Shadi Tahvildar-Zadeh,
On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., 47 (1994), 719-754.
doi: 10.1002/cpa.3160470507. |
[35] |
M. Struwe,
Radially symmetric wave maps from (1+2)-dimensional Minkowski space to the sphere, Math. Z., 242 (2002), 407-414.
doi: 10.1007/s002090100345. |
[36] |
H. J. de Vega and N. Sanchez,
Exact integrability of strings in D-dimensional de Sitter spacetime, Phys. Rev. D, 47 (1993), 3394-3404.
doi: 10.1103/PhysRevD.47.3394. |
[37] |
Z. Wang,
Symmetries and the calculations of degree, Chin. Ann. of Math. B, 10 (1989), 520-536.
|
[38] |
R. S. Ward and A. E. Winn, Integrable systems admitting topological solitons, J. Phys. A, 31 (1998), L261–L266.
doi: 10.1088/0305-4470/31/13/003. |
show all references
References:
[1] |
J. P. Antoine and B. Piette,
Solutions of Euclidean σ models on noncompact Grassmannian manifolds, J. Math. Phys., 29 (1988), 1687-1697.
doi: 10.1063/1.527917. |
[2] |
K. Arthur, D. Tchrakian and Y. Yang,
Topological and nontopological self-dual Chern-Simons solitions in a gauged O(3) σ model, Phys. Rev. D, 54 (1996), 5245-5248.
doi: 10.1103/PhysRevD.54.5245. |
[3] |
D. Bartolucci, C.-C. Chen, C. S. Lin and G. Tarantello,
Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations, 29 (2004), 1241-1265.
doi: 10.1081/PDE-200033739. |
[4] |
D. Bartolucci, Y. Lee, C. S. Lin and M. Onodera,
Asymptotic analysis of solutions to a gauged O(3) sigma model, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 32 (2015), 651-685.
doi: 10.1016/j.anihpc.2014.03.001. |
[5] |
D. Bartolucci and G. Tarantello,
Liouville Type equations with singular data and their applications to periodic multivortices for the electroweak Theory, Comm. Math. Phys., 229 (2000), 3-47.
doi: 10.1007/s002200200664. |
[6] |
A. A. Belavin and A. M. Polyakov,
Metastable states of two dimensional isotropic ferromagnets, JETP Lett., 22 (1975), 245-247.
|
[7] |
H. Brezis and J-M. Coron,
Large solutions for harmonic maps in two dimensions, Comm. Math. Phys., 92 (1983), 203-215.
doi: 10.1007/BF01210846. |
[8] |
H. Brezis and F. Merle,
Uniform estimates and blowup behavior for solutions of $-\Delta u = V(x)e^u$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.
doi: 10.1080/03605309108820797. |
[9] |
H. Chan, C.-C. Fu and C.-S. Lin,
Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equation, Comm. Math. Phys., 231 (2002), 189-221.
doi: 10.1007/s00220-002-0691-6. |
[10] |
W. Chen and C. Li,
Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[11] |
W. Chen and C. Li,
Qualitative properties of solutions to some nonlinear elliptic equations in $\mathbb{R}^2$, Duke Math. J., 71 (1993), 427-439.
doi: 10.1215/S0012-7094-93-07117-7. |
[12] |
X. Chen, S. Hastings, J. B. McLeod and Y. Yang,
A nonlinear problem elliptic equation arising from gauge field theory and cosmology, Proc. Roy. Soc. London Ser. A, 446 (1994), 453-478.
doi: 10.1098/rspa.1994.0115. |
[13] |
K. Choe, J. Han, C.-S. Lin and T.-C. Lin,
Uniqueness and solution structure of nonlinear equations arising from the Chern-Simons gauged O(3) sigma models, J. Differential Equations, 255 (2013), 2136-2166.
doi: 10.1016/j.jde.2013.06.010. |
[14] |
K. Choe, J. Han and C.-S. Lin,
Bubbling solutions for the Chern-Simons gauged O(3) sigma modelin $\mathbb{R}^2$, Discrete Contin. Dyn. Syst., 34 (2014), 2703-2728.
doi: 10.3934/dcds.2014.34.2703. |
[15] |
K. Choe, J. Han, Y. Lee and C.-S. Lin,
Bubbling solutions for the Chern-Simons gauged O(3) sigma model on a torus, Calc. Var. Partial Differential Equations, 54 (2015), 1275-1329.
doi: 10.1007/s00526-015-0825-2. |
[16] |
K. Choe, N. Kim and C.-S. Lin,
Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 837-852.
doi: 10.1016/j.anihpc.2011.06.003. |
[17] |
P. K. Ghosh and S. K. Ghosh,
Topological and nontopological solitons in a gauged O(3) sigma model with Chern-Simons term, Phys. Lett. B, 366 (1996), 199-204.
doi: 10.1016/0370-2693(95)01365-2. |
[18] |
A. C. Huang, Harmonic maps of punctured surfaces to hyperbolic plane, preprint, arXiv: 1605.07715v1. |
[19] |
H. Huh and G. Jin, Local and global solutions of Chern-Simons gauged O(3) sigma equations in one space dimension, J. Math. Phys., 57 (2016), 081511, 11pp.
doi: 10.1063/1.4960744. |
[20] |
E. Hulett,
Harmonic superconformal maps of surfaces in $\Bbb{H}^n$, J. Geo. Phys., 42 (2002), 139-165.
doi: 10.1016/S0393-0440(01)00082-1. |
[21] |
A. Jaffe and C. Taubes, Vortices and Monopoles, Birkhäuser Boston, 1980. |
[22] |
S.-S. Kim and P. Oh,
On the gauged noncompact spin system, International Journal of Modern Physics A, 13 (1998), 5503-5517.
doi: 10.1142/S0217751X9800250X. |
[23] |
K. Kimm, K. Lee and T. Lee,
Anyonic Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Rev. D, 53 (1996), 4436-4440.
|
[24] |
J. Krieger,
Global regularity of wave maps from $\mathbb{R}^{2+1}$ to $\mathbb{H}^2$. Small energy, Comm. Math. Phys., 250 (2004), 507-580.
doi: 10.1007/s00220-004-1088-5. |
[25] |
A. Kundu,
On σ-models with noncompact groups, Lett. Math. Phys., 6 (1982), 479-485.
doi: 10.1007/BF00405869. |
[26] |
A. Kundu,
Gauge equivalence of sigma models with non-compact Grassmannian manifolds, J. Phys. A: Math. Gen., 19 (1986), 1303-1314.
doi: 10.1088/0305-4470/19/8/012. |
[27] |
P. O. Mazur,
A relationship between the electrovacuum Ernst equations and nonlinear σ-model, Acta Phys. Polon. B, 14 (1983), 219-234.
|
[28] |
P. O. Mazur,
A global identity for nonlinear σ-models, Phys. Lett. A, 100 (1984), 341-344.
doi: 10.1016/0375-9601(84)91084-3. |
[29] |
R.-C. McOwen,
Conformal metrics in $\mathbb{R}^2$ with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J., 34 (1985), 97-104.
doi: 10.1512/iumj.1985.34.34005. |
[30] |
L. Nirenberg, Topics in Nonlinear Functional Analysis, American Mathematical Society, 2001.
doi: 10.1090/cln/006. |
[31] |
P. Oh,
Bogomol'nyi solitons and Hermitian symmetric spaces, Rep. Math. Phys., 43 (1999), 271-281.
doi: 10.1016/S0034-4877(99)80035-4. |
[32] |
I. Rodnianski and J. Sterbenz,
On the formation of singularities in the critical O(3) σ-model, Ann. of Math., 172 (2010), 187-242.
doi: 10.4007/annals.2010.172.187. |
[33] |
B. J. Schroers,
Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Lett. B, 356 (1995), 291-296.
doi: 10.1016/0370-2693(95)00833-7. |
[34] |
J. Shatah and A. Shadi Tahvildar-Zadeh,
On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., 47 (1994), 719-754.
doi: 10.1002/cpa.3160470507. |
[35] |
M. Struwe,
Radially symmetric wave maps from (1+2)-dimensional Minkowski space to the sphere, Math. Z., 242 (2002), 407-414.
doi: 10.1007/s002090100345. |
[36] |
H. J. de Vega and N. Sanchez,
Exact integrability of strings in D-dimensional de Sitter spacetime, Phys. Rev. D, 47 (1993), 3394-3404.
doi: 10.1103/PhysRevD.47.3394. |
[37] |
Z. Wang,
Symmetries and the calculations of degree, Chin. Ann. of Math. B, 10 (1989), 520-536.
|
[38] |
R. S. Ward and A. E. Winn, Integrable systems admitting topological solitons, J. Phys. A, 31 (1998), L261–L266.
doi: 10.1088/0305-4470/31/13/003. |
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