Nonexistence of solutions of higher-order nonlinear non-Gauge Schrödinger equation
DOI:
https://doi.org/10.5269/bspm.47911Abstract
A nonexistence result is proved of the space higher-order nonlinear Schrodinger equation
where m > 1, n > 1 and p > n. Our method of proof rests on a judicious choice of the test function in the weak formulation of the equation. Then, we obtain an upper bound of the life span of solutions. Furthermore, the necessary conditions for the existence of local or global solutions are provided.
Next, we extend our results to the 2 × 2 – system.
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3. Biswas, A., Khalique, C. M., Stationay solutions for nonlinear dispersive Schrodinger's equation, Nonlinear Dyn. 63, 623-626, (2011). https://doi.org/10.1007/s11071-010-9824-1
4. Biswas, A., Khalique, C. M., Stationary Solutions for the Nonlinear Dispersive Schrodinger Equation with Generalized Evolution, Chinese J. Phys. 51, 103-110, (2013).
5. Cazenave, T., Semilinear Schr¨odinger equations, Courant lecture Notes in Mathematics 10, American Mathematical Society (2003). https://doi.org/10.1090/cln/010
6. Chaves, M.. Galaktionov, V., Regional blow-up for a higher-order semilinear parabolic equation, Europ J. Appl. Math. 12, 601- 623, (2001). https://doi.org/10.1017/S0956792501004685
7. Galaktionov, V. A., Mitidieri, E. L., Pohozaev S. I, Blow-up for Higher-order parabolic, Hyperbolic, Dispersion and Schrodinger Equations, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 568 pp, (2014), https://doi.org/10.1201/b17415
8. Galaktionov, V., Pohozaev, S. I., Existence and blow-up for higher-order semilinear parabolic equation: majorizing order perserving operator, Indiana Univ. Math. J. 51, 2321-1338, (2002). https://doi.org/10.1512/iumj.2002.51.2131
9. Ikeda, M., Lifespan of solutions for the nonlinear Schrodinger equation without gauge invariance, arXiv:1211.6928v2 [math.AP] (2012).
10. Ikeda, M., Wakasugi, Y., Small data blow-up of L2 -solution for the nonlinear Schr¨odinger equation without gauge invariance, Differential Integral Equations 26, 1275-1285, (2013).
11. Lange, H., Peppenperg, M., Teismann, H., Nash-Moser methods for the solutions of quasilinear Schr¨odinger equations, Comm. Part. Differ. Equat., 24, 1399-1418, (1999). https://doi.org/10.1080/03605309908821469
12. Majda, A. J., McLaughlin, D. W., Tabak, E. G., A One-Dimensional Model for Dispersive Wave Turbulence, J. Nonlinear Sci. 6, 9-44, (1997). https://doi.org/10.1007/BF02679124
13. Zakharov, V., Dias, F., Pushkarev, A., One-dimensional wave turbulence, Physics Reports 398, 1-65, (2004). https://doi.org/10.1016/j.physrep.2004.04.002
14. Nobre, F. D., Rego-Monteiro, M. A., Tsallis, C., Nonlinear Relativistic and Quantum Equations with a Common Type of Solution, Physical Review Letters 106(14):140601, (2011). https://doi.org/10.1103/PhysRevLett.106.140601
15. Pohozaev, S. I., Nonexistence of global solutions of nonlinear evolution equations, Differ. Equ., 49, 599-606, (2013). https://doi.org/10.1134/S001226611305008X
16. Sulem, C., Sulem, P.L., The Nonlinear Schrodinger Equation: Self-Foscusing and Wave Collapse, Applied Mathematics Sciences, Series in Mathematical Sciences, Volume 139, Springer-Verlag, (1999).
17. Ogawa, T., Blow-up of H1 Solution for the Nonlinear Schrodinger Equation, J. Differential Equations 92, 317-330, (1991). https://doi.org/10.1016/0022-0396(91)90052-B
18. Yan, Z., Envelope compactons and solitary patterns, Phys. Lett. A 355, 212-215, (2006). https://doi.org/10.1016/j.physleta.2006.02.032
19. Z. Yan, Envelope compact and solitary pattern structures for the GNLS(m,n,p,q) equations, Phys. Lett. A 357, 196-203, (2006). https://doi.org/10.1016/j.physleta.2006.04.032
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2022-01-26
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