Polynomial complex Ginzburg-Landau equations in Zhidkov spaces

A. Besteiro

doi:10.30970/ms.52.1.55-62

We consider the so-called Complex Ginzburg-Landau equations with a polynomial nonlin- earity in the real line. We prove existence results concerned with the initial value problem for these equations in Zhidkov spaces with a new approach using Splitting methods.

1. I.S. Aranson, L. Kramer, The world of the complex Ginzburg-Landau equation, Reviews of Modern Physics, 74 (2002), №1, 99-143.

2. Z. Asgari, M. Ghaem, M.G. Mahjani, Pattern formation of the FitzHugh-Nagumo model:cellular automata approach, Iran. J. Chem. Chem. Eng., 30 (2011), №1, 135-142.

3. A.T. Besteir, D.F. Rial, Global existence for vector valued fractional reaction-diffusion equations, arXiv preprint arXiv:1805.09985, 2018.

4. D. Blackmore, A.K. Prykarpatski, Dark equations and their light integrability, J. Nonlin. Math. Phys., 21 (2014), №3, 407-428.

5. J.P. Borgna, M.De Leo, D. Rial, C. Sanchez de la Vega, General Splitting methods for abstract semilinear evolution equations, Commun. Math. Sci, Int. Press Boston, Inc., 13 (2015), 83-101.

6. T. Cazenave, A. Haraux, An introduction to semilinear evolution equations, Oxford Lecture Ser. Math. Appl., Clarendon Press, Rev ed. edition, 1999.

7. W. Cao, J. Fei, Z. Ma, Q. Liu, Bosonization and new interaction solutions for the coupled Korteweg-de Vries system, Waves in random and complex media, 2018.

8. M. De Leo, D. Rial, C. Sanchez de la Vega, High-order time-splitting methods for irreversible equations, IMA J. Numer. Anal., (2015), 1842-1866.

9. N. Efremidis, K. Hizanidis, H.E. Nistazakis, D.J. Frantzeskakis, B.A. Malomed, Stabilization of dark solitons in the cubic Ginzburg-Landau equation, Physical Review E, 63 (2000), №5, 7410-7414.

10. K.J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, Springer Science & Business Media, V.194, 1999.

11. R.A. Fisher, The wave of advance of advantage genes, Ann. Eugen., 7 (1937), №4, 353-369.

12. R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), №4, 257-278.

13. C. Gallo, Schrodinger group on Zhidkov spaces, Adv. Differential Equations, 9 (2004), №5-6, 509-538.

14. J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods, Physica D, 95 (1996), №3-4, 191-228.

15. J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Contraction methods, Comm. Math. Phys., 187 (1997), №1, 45-79.

16. Y.S. Kivshar, B. Luther-Davies, Dark optical solitons: physics and applications, Physics reports, 298 (1998), №2.3, 81-197.

17. A. Kolmogorov, I. Petrovsy, N. Piskounov, Study of the diffusion equation with growth of the quantity of matter and its applications to a biological problem, Moscow Univ. Math. Bull., 1989, 105 p.

18. B.A. Kupershmidt, Dark equations, J. Nonlin. Math. Phys., 8 (2001), №3, 363-445.

19. P. Zhidkov, The Cauchy problem for a nonlinear Schrodinger equation, Joint Inst. for Nuclear Research, JINR-R.5-87-373, 1987.

	Polynomial complex Ginzburg-Landau equations in Zhidkov spaces
Author	A. Besteiro abesteiro@dm.uba.ar Instituto de Matematica Luis Santalo, CONICET-UBA Ciudad Universitaria, Pabellon I Buenos Aires, Argentina
Abstract	We consider the so-called Complex Ginzburg-Landau equations with a polynomial nonlin- earity in the real line. We prove existence results concerned with the initial value problem for these equations in Zhidkov spaces with a new approach using Splitting methods.
Keywords	well-posedness; Zhidkov spaces; Lie–Trotter method
DOI	doi:10.30970/ms.52.1.55-62
Reference	1. I.S. Aranson, L. Kramer, The world of the complex Ginzburg-Landau equation, Reviews of Modern Physics, 74 (2002), №1, 99-143. 2. Z. Asgari, M. Ghaem, M.G. Mahjani, Pattern formation of the FitzHugh-Nagumo model:cellular automata approach, Iran. J. Chem. Chem. Eng., 30 (2011), №1, 135-142. 3. A.T. Besteir, D.F. Rial, Global existence for vector valued fractional reaction-diffusion equations, arXiv preprint arXiv:1805.09985, 2018. 4. D. Blackmore, A.K. Prykarpatski, Dark equations and their light integrability, J. Nonlin. Math. Phys., 21 (2014), №3, 407-428. 5. J.P. Borgna, M.De Leo, D. Rial, C. Sanchez de la Vega, General Splitting methods for abstract semilinear evolution equations, Commun. Math. Sci, Int. Press Boston, Inc., 13 (2015), 83-101. 6. T. Cazenave, A. Haraux, An introduction to semilinear evolution equations, Oxford Lecture Ser. Math. Appl., Clarendon Press, Rev ed. edition, 1999. 7. W. Cao, J. Fei, Z. Ma, Q. Liu, Bosonization and new interaction solutions for the coupled Korteweg-de Vries system, Waves in random and complex media, 2018. 8. M. De Leo, D. Rial, C. Sanchez de la Vega, High-order time-splitting methods for irreversible equations, IMA J. Numer. Anal., (2015), 1842-1866. 9. N. Efremidis, K. Hizanidis, H.E. Nistazakis, D.J. Frantzeskakis, B.A. Malomed, Stabilization of dark solitons in the cubic Ginzburg-Landau equation, Physical Review E, 63 (2000), №5, 7410-7414. 10. K.J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, Springer Science & Business Media, V.194, 1999. 11. R.A. Fisher, The wave of advance of advantage genes, Ann. Eugen., 7 (1937), №4, 353-369. 12. R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), №4, 257-278. 13. C. Gallo, Schrodinger group on Zhidkov spaces, Adv. Differential Equations, 9 (2004), №5-6, 509-538. 14. J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods, Physica D, 95 (1996), №3-4, 191-228. 15. J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Contraction methods, Comm. Math. Phys., 187 (1997), №1, 45-79. 16. Y.S. Kivshar, B. Luther-Davies, Dark optical solitons: physics and applications, Physics reports, 298 (1998), №2.3, 81-197. 17. A. Kolmogorov, I. Petrovsy, N. Piskounov, Study of the diffusion equation with growth of the quantity of matter and its applications to a biological problem, Moscow Univ. Math. Bull., 1989, 105 p. 18. B.A. Kupershmidt, Dark equations, J. Nonlin. Math. Phys., 8 (2001), №3, 363-445. 19. P. Zhidkov, The Cauchy problem for a nonlinear Schrodinger equation, Joint Inst. for Nuclear Research, JINR-R.5-87-373, 1987.
Pages	55-62
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
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Table of content of issue	html