The cauchy problem for coupled nonlinear Schrödinger equations with linear damping: Local and global existence and blowup of solutions

Abstract

The authors study, by applying and extending the methods developed by Cazenave (2003), Dias and Figueira (2014), Dias et al. (2014), Glassey (1994–1997), Kato (1987), Ohta and Todorova (2009) and Tsutsumi (1984), the Cauchy problem for a damped coupled system of nonlinear Schrödinger equations and they obtain new results on the local and global existence of H ¹-strong solutions and on their possible blowup in the supercritical case and in a special situation, in the critical or supercritical cases.

Keywords

Nonlinear Schrödinger equations / Cauchy problem / Blowup of solutions / Dissipation

References

[1] Bludov Y.u. V., Driben R., Konotop V. V., Malomed B. A.. Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides. J. Opt., 2013, 15: 064010

[2] Cazenave T.. Semilinear Schrödinger Equations, 2003

[3] Couairon A., Mysyrowicz A.. Femtosecond Filamentation in Transparent Media. Phys. Rep., 2007, 441: 47-189

[4] Dias J. P., Figueira M.. On the blowup of solutions of a Schrödinger equation with an inhomogeneous damping coefficient. Comm. Contemp. Math., 2014, 16: 1350036

[5] Dias J. P., Figueira M., Konotop V. V., Zezyulin D. A.. Supercritical blowup in coupled paritytime-symmetric nonlinear Schrödinger equations. Studies Appl. Math., 2014, 133: 422-440

[6] Glassey R. T.. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys., 1977, 18: 1794-1797

[7] Jüngel A., Weishäupl R. M.. Blow-up in two-component nonlinear Schrödinger systems with an external driven field. Math. Models Meth. Appl. Sciences, 2013, 23: 1699-1727

[8] Kato T.. On nonlinear Schrödinger equations. Ann. Inst. H. PoinCaré Phys. Théor., 1987, 46: 113-129

[9] Menyuk C. R.. Pulse propagation in an elliptically birefringent medium. IEEE J. Quant. Electron., 1989, 25: 2674

[10] Ohta M., Todorova G.. Remarks on global existence and blowup for damped nonlinear Schrödinger equations. Discrete Cont. Dyn. Syst., 2009, 23: 1313-1325

[11] Pitaevskii L., Stringari S.. Bose–Einstein Condensation, 2003, Oxford: Clarendon Press

[12] Prytula V., Vekslerchik V., Pérez-Garcia V. M.. Collapse in coupled nonlinear Schrödinger equations: Sufficient conditions and applications. Physica D, 2009, 238: 1462-1467

[13] Roberts D. C., Newell A. C.. Finite-time collapse of N classical fields described by coupled nonlinear Schrödinger equations. Phys. Rev. E, 2006, 74: 047602

[14] Tsutsumi M.. Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations. SIAM J. Math. Anal., 1984, 15: 357-366