\(H^1\)-scattering for systems of \(N\)-defocusing weakly coupled NLS equations in low space dimensions. (English) Zbl 1318.35103
Summary: We prove that the scattering operators and wave operators are well-defined in the energy space for the system of defocusing Schrödinger equations
\[
\begin{cases} i \partial_t u_\mu +\Delta u_\mu - \sum\limits_{\mu, \nu = 1}^N \beta_{\mu \nu}| u_\nu|^{p+1}| u_\mu|^{p-1} u_\mu = 0,\quad \mu = 1, \dots, N, \\ (u_\mu(0, \cdot))_{\mu = 1}^N = (u_{\mu, 0})_{\mu = 1}^N \in H^1(\mathbb{R}^d)^N, \end{cases}
\]
with \(N \geq 2\), \(\beta_{\mu \nu} \geq 0\), \(\beta_{\mu \mu} \neq 0\) for \(p > 2\) if \(d = 1\), \(p > 1\) if \(d = 2\) and \(1 \leq p < 2\) if \(d = 3\).
References:
| [1] | Adams, R.; Fournier, J., Sobolev Spaces (2003), Academic Press · Zbl 1098.46001 |
| [2] | Cazenave, T., Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10 (2003), New York University Courant Institute of Mathematical Sciences: New York University Courant Institute of Mathematical Sciences New York · Zbl 1055.35003 |
| [3] | Colliander, J.; Grillakis, M.; Tzirakis, N., Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 62, 7, 920-968 (2009) · Zbl 1185.35250 |
| [4] | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Scattering for the 3D cubic NLS below the energy norm, Comm. Pure Appl. Math., 57, 987-1014 (2004) · Zbl 1060.35131 |
| [5] | Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Global well-posedness and scattering in the energy space for the critical nonlinear Schrödinger equation in \(R^3\), Ann. of Math. (2), 167, 3, 767-865 (May 2008) · Zbl 1178.35345 |
| [6] | Colorado, E., Positive solutions to some systems of coupled nonlinear Schrödinger equations, preprint · Zbl 1307.35275 |
| [7] | D’Ancona, P.; Fanelli, L.; Vega, L.; Visciglia, N., Endpoint Strichartz estimates for the magnetic Schrödinger equation, J. Funct. Anal., 258, 3227-3240 (2010) · Zbl 1188.81061 |
| [8] | Fanelli, L.; Lucente, S.; Montefusco, E., Semilinear Hamiltonian Schrödinger systems, Int. J. Dyn. Syst. Differ. Equ., 3, 401-422 (2011) · Zbl 1234.35245 |
| [9] | Fanelli, L.; Montefusco, E., On the blow-up threshold for weakly coupled nonlinear Schrödinger equations, J. Phys. A. J. Phys. A, Math. Ann., 344, 2, 249-278 (2009) |
| [10] | Fanelli, L.; Vega, L., Magnetic virial identities, weak dispersion and Strichartz estimates, Math. Ann., 344, 249-278 (2009) · Zbl 1163.35005 |
| [11] | Garcia, A., Magnetic virial identities and applications to blow-up for Schrödinger and wave equations, J. Phys. A, 45, 1, 015202 (2012), 16 pp · Zbl 1234.35045 |
| [12] | Ginibre, J.; Velo, G., Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl., 64, 363-401 (1985) · Zbl 0535.35069 |
| [13] | Ginibre, J.; Velo, G., Quadratic Morawetz inequalities and asymptotic completeness in the energy space e for nonlinear Schrödinger and Hartree equations, Quart. Appl. Math., 68, 113-134 (2010) · Zbl 1186.35201 |
| [14] | Gonçalves Ribeiro, J. M., Finite time blow-up for some nonlinear Schrödinger equations with an external magnetic field, Nonlinear Anal., 16, 11, 941-948 (1991) · Zbl 0734.35127 |
| [15] | Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities, 324 (1952), Cambridge University Press · Zbl 0634.26008 |
| [16] | Keel, M.; Tao, T., Endpoint Strichartz estimates, Amer. J. Math., 120, 5, 955-980 (1998) · Zbl 0922.35028 |
| [17] | Lin, J.; Strauss, W., Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30, 245-263 (1978) · Zbl 0395.35070 |
| [18] | Lin, T. C.; Wei, J., Ground state of \(N\) coupled nonlinear Schrödinger equations in \(R^n, n \leq 3\), Comm. Math. Phys., 255, 629-653 (2005) · Zbl 1119.35087 |
| [19] | Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 2, 109-145 (1984) · Zbl 0541.49009 |
| [20] | Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 4, 223-283 (1984) · Zbl 0704.49004 |
| [21] | Maia, L. A.; Montefusco, E.; Pellacci, B., Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229, 743-767 (2006) · Zbl 1104.35053 |
| [22] | Morawetz, C., Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. Sect. A, 206, 291-296 (1968) · Zbl 0157.41502 |
| [23] | Nakanishi, K., Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Fund. Anal., 169, 201-225 (1999) · Zbl 0942.35159 |
| [24] | Nakanishi, K., Remarks on the energy scattering for nonlinear Klein-Gordon and Schrödinger equations, Tohoku Math. J. (2), 53, 2, 171-336 (2001) |
| [25] | Nguyen, N. V.; Tian, R.; Deconinck, B.; Sheils, N., Global existence for a coupled system of Schrödinger equations with power-type nonlinearities, J. Math. Phys., 54, 011503 (2013) · Zbl 1286.35230 |
| [26] | Planchon, F.; Vega, L., Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42, 2, 261-290 (2009) · Zbl 1192.35166 |
| [27] | Pomponio, A., Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227, 1, 258-281 (2006) · Zbl 1100.35098 |
| [28] | Terracini, S.; Tzvetkov, N.; Visciglia, N., The Nonlinear Schrödinger equation ground states on product spaces, Anal. PDE, 7, 1, 73-96 (2014) · Zbl 1294.35148 |
| [29] | Tzvetkov, N.; Visciglia, N., Well-posedness and scattering for NLS on \(R^d \times T\) in the energy space, to appear in Rev. Mat. Iberoam. · Zbl 1365.35164 |
| [30] | Visciglia, N., On the decay of solutions to a class of defocusing NLS, Math. Res. Lett., 16, 5, 919-926 (2009) · Zbl 1194.35431 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
