Vectorial approach to coupled nonlinear Schrödinger systems under nonlocal Cauchy conditions

3 years ago

Abstract

The paper presents a vectorial approach for coupled general nonlinear Schrödinger systems with nonlocal Cauchy conditions. Based on fixed-point principles, the use of matrices with spectral radius less than one, and on basic properties of the Schrödinger solution operator, several existence results are obtained. The essential role of the support of the nonlocal Cauchy condition is emphasized and fully exploited.

Authors

Renata Bunoiu
Department of Mathematics, Babeş-Bolyai University, Cluj, Romania.

Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

Keywords

nonlinear Schrödinger equation; nonlocal condition; coupled system; global existence; nonlinear operator; fixed point

Paper coordinates

R. Precup, R. Bunoiu, Vectorial approach to coupled nonlinear Schrödinger systems under nonlocal Cauchy conditions, Appl. Anal. 95 (2016), 731-747, https://doi.org/10.1080/00036811.2015.1028921

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Journal

Applicable Analysis

Publisher Name

DOI

https://doi.org/10.1080/00036811.2015.1028921

Print ISSN

1563-504X

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[1] Akhmediev N, Ankiewicz A. Solitons, Nonlinear pulses and beams. London: Champman & Hall; 1997. [Google Scholar]
[2] Hajaiej H. Schrödinger systems arising in nonlinear optics and quantum mechanics: Part I. Math. Models Methods Appl. Sci. 2012;22:1250010, 1–27. [Crossref], [Web of Science ®], [Google Scholar]
[3] Ambrosetti A, Colorado E, Ruiz D. Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations. Calc. Var. Par. Differ. Equ. 2007;30:85–112. [Crossref], [Web of Science ®], [Google Scholar]
[4] Antoine X, Duboscq R. Modeling and computation of Bose–Einstein condensates: stationary states, nucleation, dynamics, stochasticity. Lectures notes in mathematics/physics. Springer. Forthcoming. [Google Scholar]
[5] Antonelli P, Weishäupl RM. Asymptotic behavior of nonlinear Schrödinger systems with linear coupling. J. Hyper. Differ. Equ. 2014;11:159. doi:10.1142/S0219891614500040. [Crossref], [Web of Science ®], [Google Scholar]
[6] Bartsch T, Wang ZQ. Note on ground states of nonlinear Schrödinger systems. J. Par. Differ. Equ. 2006;19:200–207. [Google Scholar]
[7] Bartsch T, Wang ZQ, Wei J. Bound states for a coupled Schrödinger system. J. Fixed Point Theory Appl. 2007;2:353–367. [Crossref], [Web of Science ®], [Google Scholar]
[8] Carles R. Semi-classical analysis for nonlinear Schrödinger equations. Singapore: World Scientific; 2008. [Crossref], [Google Scholar]
[9] Dancer EN, Wei J, Weth T. A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Ann. Inst. H. Poincaré Anal. Non Linéaire. 2010;27:953–969. [Crossref], [Web of Science ®], [Google Scholar]
[10] Jüngel A, Weishäupl RM. Blow-up in two-component nonlinear Schrödinger systems with an external driven field. Math. Models Methods Appl. Sci. 2013;23:1699–1727. [Crossref], [Web of Science ®], [Google Scholar]
[11] Lin TC, Wei J. Ground state of N coupled nonlinear Schrödinger equations in Rⁿ, n ≤ 3. Commun. Math. Phys. 2005;255:629–653. [Crossref], [Web of Science ®], [Google Scholar]
[12] Nguyen NV, Tian R, Deconinck B, Sheils N. Global existence for a coupled system of Schrödinger equations with power-type nonlinearities. J. Math. Phys. 2013;54:011503, 1–19. [Crossref], [Web of Science ®], [Google Scholar]
[13] Pomponio A. Coupled nonlinear Schrödinger systems with potentials. J. Differ. Equ. 2006;227:258–281. [Crossref], [Web of Science ®], [Google Scholar]
[14] Sirakov B. Least energy solitary waves for a system of nonlinear Schrödinger equations in Rⁿ. Commun. Math. Phys. 2007;271:199–221. [Crossref], [Web of Science ®], [Google Scholar]
[15] Maia LA, Montefusco E, Pellacci B. Weakly coupled nonlinear Schrödinger systems: the saturation effect. Calc. Var. Par. Differ. Equ. 2013;46:325–351. [Crossref], [Web of Science ®], [Google Scholar]
[16] Cazenave T. Semilinear Schrödinger equations. Vol. 10, Courant lecture notes in mathematics. Providence (RI): American Mathematical Society; 2003. [Crossref], [Google Scholar]
[17] Bolojan-Nica O, Infante G, Precup R. Existence results for systems with coupled nonlocal conditions. Nonlinear Anal. 2014;94:231–242. [Crossref], [Web of Science ®], [Google Scholar]
[18] Byszewski L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 1991;162:494–505. [Crossref], [Web of Science ®], [Google Scholar]
[19] Byszewski L, Lakshmikantham V. Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 1990;40:11–19. [Taylor & Francis Online], [Google Scholar]
[20] Xue X. Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces. Electron. J. Differ. Equ. 2005;64:1–7. [Google Scholar]
[21] Ashyralyev A, Sirma A. Nonlocal boundary value problems for the Schrödinger equation. Comput. Math. Appl. 2008;55:392–407. [Crossref], [Web of Science ®], [Google Scholar]
[22] Perov AI. On the Cauchy problem for a system of ordinary differential equations. Pviblizhen. Met. Reshen. Differ. Uvavn. 1964;2:115–134. Russian. [Google Scholar]
[23] Precup R. The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comp. Modell. 2009;49:703–708. [Crossref], [Web of Science ®], [Google Scholar]
[24] Precup R, Trif D. Multiple positive solutions of non-local initial value problems for first order differential systems. Nonlinear Anal. 2012;75:5961–5970. [Crossref], [Web of Science ®], [Google Scholar]
[25] Precup R. Linear and semilinear partial differential equations. Berlin: De Gruyter; 2013. [Google Scholar]
[26] Lions JL, Magenes E. Problèmes aux limites non homogènes et applications [Nonhomogeneous boundary value problems and applications]. Paris: Dunod; 1968. [Google Scholar]
[27] Varga RS. Matrix iterative analysis. 2nd ed. Berlin: Springer; 2000. [Crossref], [Google Scholar]
[28] Bunoiu R, Cardone G, Suslina T. Spectral approach to homogenization of an elliptic operator periodic in some directions. Math. Methods Appl. Sci. 2011;34:1075–1096. [Crossref], [Web of Science ®], [Google Scholar]