Exact solutions of a two-dimensional nonlinear Schrödinger equation. (English) Zbl 1241.35191

Summary: We convert the resulting nonlinear equation for the evolution of weakly nonlinear hydrodynamic disturbances on a static cosmological background with self-focusing in a two-dimensional nonlinear Schrödinger (NLS) equation. Applying the function transformation method, the NLS equation is transformed to an ordinary differential equation, which depends only on one function \(\xi \) and can be solved. The general solution of the latter equation in \(\zeta \) leads to a general solution of the NLS equation. A new set of exact solutions for the two-dimensional NLS equation is obtained.


35Q55 NLS equations (nonlinear Schrödinger equations)
35A24 Methods of ordinary differential equations applied to PDEs
35Q85 PDEs in connection with astronomy and astrophysics
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[1] Ivanov, R. R., Nuclear Phys. B, 694, 509 (2004)
[2] Khater, A. H.; Helal, M. A.; Seadawy, A. R., IL Nuovo Cimento, 115B, 1303 (2000)
[3] Khater, A. H.; Callebaut, D. K.; Seadawy, A. R., Phys. Scr., 62, 353 (2000)
[4] Klaus, M.; Shaw, J. K., Opt. Commun., 197, 491 (2001)
[5] Kates, R. E.; Kaup, D. J., Astron. Astrophys., 206, 9 (1988)
[6] Kates, R. E., Astron. Astrophys., 168, 1 (1986)
[7] Khater, A. H.; Callebaut, D. K.; Seadawy, A. R., Phys. Scr., 67, 340 (2003)
[8] Agrawal, G. P., Nonlinear Fiber Optics (1989), Academic: Academic San Diego
[9] Zakharov, V. E.; Shabat, A. B., Zh. Eksp. Teor. Fiz.. Zh. Eksp. Teor. Fiz., Sov. Phys. JETP, 34, 62 (1972)
[10] Hasegawa, A.; Tappert, F., Appl. Phys. Lett., 23, 142 (1973)
[11] Freitas, D. S.; De Oliveira, J. R., Modern Phys. Lett. B, 16, 27 (2002)
[12] Palacios, S. L., J. Nonlinear Opt. Phys. Mater., 10, 293 (2001)
[13] Zheng, C. L.; Sheng, Z. M., Int. J. Mod. Phys. B, 17, 4407 (2003)
[14] Wang, C.; Theocharis, G.; Kevrekidis, P. G.; Whitaker, N.; Law, K. J.; Frantzeskakis, D. J.; Malomed, B. A., Phys. Rev. E, 80, 4 (2009)
[15] Biswas, A., Int. J. Theor. Phys., 48, 689 (2008)
[16] Chen, Q. Y.; Kevrekidis, P. G.; Malomed, B. A., Eur. Phys. J. D, 58, 141 (2010)
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