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2024 | OriginalPaper | Buchkapitel

An Application of the Simplest Equations Method to Logarithmic Schrödinger Equation

verfasst von : Ivan P. Jordanov

Erschienen in: New Trends in the Applications of Differential Equations in Sciences

Verlag: Springer Nature Switzerland

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Abstract

Das Kapitel befasst sich mit der Anwendung der Simple Equations Method (SEsM) zur Lösung der logarithmischen Schrödingergleichung, einem entscheidenden Werkzeug in verschiedenen wissenschaftlichen Disziplinen. Indem sie die Methodik in klare Schritte unterteilen, zeigen die Autoren, wie SEsM komplexe nichtlineare partielle Differentialgleichungen in leichter handhabbare Formen verwandeln kann. Die Anwendung der logarithmischen Schrödingergleichung dient als überzeugendes Beispiel, um die Wirksamkeit der Methode bei der Ableitung genauer analytischer Lösungen zu veranschaulichen. Das Kapitel bietet nicht nur einen umfassenden Leitfaden für die SEsM, sondern unterstreicht auch ihr Potenzial für die weitere Erforschung nichtlinearer Schrödingergleichungen, was sie zu einer unschätzbaren Ressource für Spezialisten auf diesem Gebiet macht.

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Vorheriges Kapitel Area and Perimeter Full Distribution Functions for Planar Poisson Line Processes and Voronoi Diagrams

Nächstes Kapitel Explicit Solutions of the Nonlinear Schrödinger-Type Equation

P. G. Drazin: Nonlinear Systems, Cambridge University Press, Cambridge, UK, (1992)

N. K. Vitanov, K. Sakai, I. P. Jordanov, S. Managi, K. Demura: Analysis of a Japan government intervention on the domestic agriculture market, Physica A, 382, 330 – 335 (2006). https://doi.org/10.1016/j.physa.2007.02.025

R. M. May, S. A. Levin, G. Sugihara. Complex systems: ecology for bankers, Nature, 451, 893 – 895 (2008). https://doi.org/10.1038/451893a

N. K. Vitanov: Science Dynamics and Research Production. Indicators, Indexes, Statistical Laws and Mathematical Models, Springer, Cham (2016).

M. Bahrami, N. Chinichian, A. Hosseiny, G. Jafari, M. Ausloos: Optimization of the post-crisis recovery plans in scale-free networks, Physica A, 540, 123203 (2020). https://doi.org/10.1016/j.physa.2019.123203

N.K. Vitanov, F. H. Busse: Upper bounds on heat transport in a horizontal fluid layer with stress-free boundaries, ZAMP, 48, Birkhäuser Verlag, Basel, 310 – 324 (1997). https://doi.org/10.1007/PL00001478

R. J. Lempert: A new decision sciences for complex systems, PNAS USA, 99, Suppl. 3, 7309 - 7313 (2002). https://doi.org/10.1073/pnas.082081699

E. V. Nikolova, N. K. Vitanov: On the Possibility of Chaos in a Generalized Model of Three Interacting Sectors, Entropy, 22(12), 1388 (2020). https://doi.org/10.3390/e22121388

N. K. Vitanov, M. Ausloos, G. Rotundo: Discrete model of ideological struggle accounting fot migration, Advances in Complex Systems, 15, Supp. 01, 1250049 (2012). https://doi.org/10.1142/S021952591250049X

10.

J. Foster: From simplistic to complex systems in economics, Cambridge Journal of Economics, 29, 873 - 892 (2005). https://doi.org/10.1093/cje/bei083

11.

N.K. Vitanov: Upper bound on the heat transport in a horizontal fluid layer of infinite Prandtl number, Physics Letters A, 248, 338-346, (1998). https://doi.org/10.1016/S0375-9601(98)00674-4

12.

N. K.Vitanov, K. N. Vitanov: Box model of migration channels, Mathematical Social Sciences, 80, 108 – 114 (2016). https://doi.org/10.1016/j.mathsocsci.2016.02.001

13.

L. Cameron, D. Larsen-Freeman: Complex Systems and Applied Linguistics, Journal of Applied Linguistics, 17, 226 - 239 (2007). https://doi.org/10.1111/j.1473-4192.2007.00148.x

14.

L. A. N. Amaral, A. Scala, M. Barthelemy, H. E. Stanley: Classes of Small-World Networks, Proceedings of the National Academy of Sciences, 97(21), 11149 – 11152 (2000). https://doi.org/10.1073/pnas.200327197

15.

N. K. Vitanov, K. N. Vitanov, H. Kantz: On the Motion of Substance in a Channel of a Network: Extended Model and New Classes of Probability Distributions, Entropy, 22, 1240 (2020). https://doi.org/10.3390/e22111240

16.

H. Kantz, T. Schreiber: Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, UK (2004)

17.

A. S. Pikovsky, D. L. Shepelyansky: Destruction of Anderson Localization by a Weak Nonlinearity, Phys. Rev. Lett., 100, American Physical Society, 094101 (2008). https://doi.org/10.1103/PhysRevLett.100.094101

18.

T. Boeck, N. K. Vitanov: Low-dimensional chaos in zero-Prandtl-number Bénard-Marangoni convection, Physical Review E, 65, 037203 (2002). https://doi.org/10.1103/PhysRevE.65.037203

19.

Z. I. Dimitrova: Numerical Investigation Of Nonlinear Waves Connected To Blood Flow In An Elastic Tube With Variable Radius, Journal of Theoretical and Applied Mechanics 45, No. 4, 79 – 92 (2015). https://doi.org/10.1515/jtam-2015-0025

20.

Y. Niu, S. Gong: Enhancing Kerr nonlinearity via spontaneously generated coherence, Phys. Rev. A, 73, 053811 (2006). https://doi.org/10.1103/PhysRevA.73.053811

21.

N. K. Vitanov: Resulat connected to time series analysis and machine learning, Studies in Computational Intelligence, 934, 363 - 383 (2021). https://doi.org/10.1007/978-3-030-72284-5_17

22.

R. Struble: Nonlinear Differential Equations, Dover, New York (2018)

23.

N. K. Vitanov, M. Ausloos. Knowledge epidemics and population dynamics models for describing idea diffusion, in Models of Science Dynamics, edited by A.Scharnhorst, K Boerner, P. van den Besselaar, Springer, Berin, p.p. 65 – 129, (2012). https://doi.org/10.48550/arXiv.1201.0676

24.

P. J. Brockwell, R. A. Davis, M. V. Calder: Introduction to Time Series and Forecasting, Springer, New York, (2002)

25.

H. Kantz, D. Holstein, M. Ragwitz, N. K. Vitanov: Markov chain model for turbulent wind speed data, Physica A, 342, 315 – 321 (2004). https://doi.org/10.1016/j.physa.2004.01.070

26.

A. B. Chambel: Applied chaos theory: A paradigm for complexity, Academic Press, Boston (1993)

27.

K. T. Ashenfelter, S. M. Boker, J. R Waddell, N. Vitanov: Spatiotemporal symmetry and multifractal structure of head movements during dyadic conversation, Journal of Experimental Psychology: Human Perception and Performance, 35, 1072 – 1091 (2009). https://doi.org/10.1037/a0015017

28.

N. K. Vitanov: Upper bounds on the heat transport in a porous layer, Physica D, 136, 322 – 339 (2000). https://doi.org/10.1016/S0167-2789(99)00165-7

29.

S. Grossberg: Nonlinear neural networks: Principles, mechanisms, and architectures, Neural Networks, 1, 17-61 (1988). https://doi.org/10.1016/0893-6080(88)90021-4

30.

N. K. Vitanov, N. Hoffmann, B. Wernitz: Nonlinear time series analysis of vibration data from a friction brake: SSA, PCA, and MFDFA, Chaos Solitons and Fractals, 69, 90 – 99 (2014). https://doi.org/10.1016/j.chaos.2014.09.010

31.

S. Goldstein: Social Psychology and Nonlinear Dynamical Systems Theory, Phychological Inquiry, 8 , 125 - 128 (1997). https://doi.org/10.1207/s15327965pli0802_6

32.

N. K. Vitanov, Z. Dimitrova, H. Kantz: On the trap of extinction and its elimination, Physics Letters A, 346, 350-355 (2006). https://doi.org/10.1016/j.physleta.2005.09.043

33.

N. K. Vitanov, K. N. Vitanov: Statistical distributions connected to motion of substance in a channel of a network, Physica A, 527, 121174 (2019). https://doi.org/10.1016/j.physa.2019.121174

34.

R. Borisov, Z. I. Dimitrova, N. K. Vitanov: Statistical Characteristics of Stationary Flow of Substance in a Network Channel Containing Arbitrary Number of Arms, Entropy, 22, 553 (2020). https://doi.org/10.3390/e22050553

35.

N. K. Vitanov, R. Borisov, K. N. Vitanov: On the motion of substance in a channel and growth of random networks, Physica A, 581, 126207 (2021). https://doi.org/10.1016/j.physa.2021.126207

36.

S. Goldstein: Social Psychology and Nonlinear Dynamical Systems Theory, Phychological Inquiry, 8, 125 - 128 (1997). https://doi.org/10.1207/s15327965pli0802_6

37.

L. Hammyl, N. Gilbert, Agent-based models in economics. (Wiley, Chichester, UK, 2016).

38.

I. P. Jordanov: On the nonlinear waves in (2+ 1)-dimensional population systems, Comptes rendus de l’Academie Bulgare des sciences 61, 307–314 (2008)

39.

I. P. Jordanov: Nonlinear waves caused by diffusion of population members, Comptes rendus de l’Academie Bulgare des sciences 62, 33–40 (2009)

40.

I. P. Jordanov: Coupled Kink Population Waves, Journal of Theoretical and Applied Mechanics, 40, n.2,93-98 (2010)

41.

I. P. Jordanov: Simple equations method applied to equations of nonlinear Schrödinger kind, AIP Conference Proceedings, 2459, 030016 (2022). https://doi.org/10.1063/5.0084340

42.

I. P. Jordanov, Z. I. Dimitrova: On Nonlinear Waves of Migration, Journal of Theoretical and Applied Mechanics, 40, n.1 89–96 (2010)

43.

I. P. Jordanov, E. V. Nikolova: On nonlinear waves in the spatio-temporal dynamics of interacting populations, Journal of Theoretical and Applied Mechanics, 43, 69–76 (2013). https://doi.org/10.2478/jtam-2013-0015

44.

I. N. Dushkov, I. P. Jordanov, N. K. Vitanov: Numerical modeling of dynamics of a population system with time delay, Mathematical Methods in the Applied Sciences, 41, 8377 – 8384 (2017). https://doi.org/10.1002/mma.4553

45.

N. K. Vitanov, I. P. Jordanov, Z. I. Dimitrova: On nonlinear population waves, Applied Mathematics and Computation, 215, 2950–2964 (2009). https://doi.org/10.1016/j.amc.2009.09.041

46.

N. K. Vitanov, I .P. Jordanov, Z. I. Dimitrova: On nonlinear dynamics of interacting populations: Coupled kink waves in a system of two populations, Communications in Nonlinear Science and Numerical Simulation, 14, 2379–2388 (2009). https://doi.org/10.1016/j.cnsns.2008.07.015

47.

N. Martinov, N. Vitanov: On the correspondence between the self-consistent 2D Poisson-Boltzmann structures and the sine-Gordon waves, Journal of Physics A: Mathematical and General, 25, L51 – L56 (1992). https://doi.org/10.1088/0305-4470/25/2/004

48.

N. Martinov, N. Vitanov: On some solutions of the two-dimensional sine-Gordon equation, Journal of Physics A: Mathematical and General, 25, L419 – L426 (1992). https://doi.org/10.1088/0305-4470/25/8/007

49.

N. Martinov, N. Vitanov: Running wave solutions of the two-dimensional sine-Gordon equation, J. Phys A: Math. Gen., 25(12), 3609 – 3613 (1992). https://doi.org/10.1088/0305-4470/25/12/021

50.

N. K. Martinov, N. K. Vitanov: New class of running-wave solutions of the (2+1)-dimensional sine-Gordon equation, Journal of Physics A: Mathematical and General, 27, 4611 – 4618 (1994). https://doi.org/10.1088/0305-4470/27/13/034

51.

N. K. Martinov, N. K. Vitanov: On self-consistent thermal equilibrium structures in two-dimensional negative-temperature systems, Canadian Journal of Physics, 72, 618 – 624 (1994). https://doi.org/10.1139/p94-079

52.

N. Vitanov, N. Martinov, : On the solitary waves in the sine-Gordon model of the two-dimensional Josephson junction, Zeitschrift für Physik B, 100, 129 – 135 (1996). https://doi.org/10.1007/s002570050102

53.

N. K. Vitanov: On travelling waves and double-periodic structures in two-dimensional sine-Gordon systems, Journal of Physics A: Mathematical and General, 29, 5195 – 5207 (1996). https://doi.org/10.1088/0305-4470/29/16/036

54.

N. K. Vitanov: Complicated exact solutions to the 2+ 1-dimensional sine-Gordon equation, ZAMM, 78, S787 - S788 (1998)

55.

N. K. Vitanov: Breather and soliton wave families for the sine-Gordon equation, Proc. Roy. Soc. London A, 454, 2409 – 2423 (1998). https://doi.org/10.1098/rspa.1998.0264

56.

N. K. Vitanov: Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity, Communications in Nonlinear Science and Numerical Simulation, 15, 2050 – 2060 (2010). https://doi.org/10.1016/j.cnsns.2009.08.011

57.

N. K. Vitanov, Z. I. Dimitrova: Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics, Communications in Nonlinear Science and Numerical Simulation 15, 2836 – 2845 (2010). https://doi.org/10.1016/j.cnsns.2009.11.029

58.

N. K. Vitanov, Z. I. Dimitrova, H. Kantz: Modified method of simplest equation and its application to nonlinear PDEs, Applied Mathematics and Computation 216, 2587 - 2595, 216, 2587 – 2595 (2010). https://doi.org/10.1016/j.amc.2010.03.102

59.

N. K. Vitanov: Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs, Communications in Nonlinear Science and Numerical Simulation, 16, 1176 – 1185 (2011). https://doi.org/10.1016/j.cnsns.2010.06.011

60.

N. K. Vitanov, Z. I. Dimitrova, K. N. Vitanov: On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis-Processi equation and b-equation, Communications in Nonlinear Science and Numerical Simulation, 16, 3033 – 3044 (2011). https://doi.org/10.1016/j.cnsns.2010.11.013

61.

N. K. Vitanov: On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: the role of the simplest equation, Communications in Nonlinear Science and Numerical Simulation, 16, 4215 – 4231 (2011). https://doi.org/10.1016/j.cnsns.2011.03.035

62.

N. K. Vitanov: On modified method of simplest equation for obtaining exact solutions of nonlinear PDEs: case of elliptic simplest equation, Pliska Stud. Math. Bulgar., 21, 257 – 266 (2012)

63.

N. K. Vitanov, Z. I. Dimitrova, H. Kantz: Application of the method of simplest equation for obtaining exact traveling-wave solutions for the extended Korteweg-de Vries equation and generalized Camassa-Holm equation, Applied Mathematics and Computation, 219, 7480 – 7492 (2013). https://doi.org/10.1016/j.amc.2013.01.035

64.

N. K. Vitanov, Z. I. Dimitrova, K. N. Vitanov: Traveling waves and statistical distributions connected to systems of interacting populations, Computers & Mathematics with Applications, 66, 1666 – 1684 (2013). https://doi.org/10.1016/j.camwa.2013.04.002

65.

N. K. Vitanov, Z. I. Dimitrova: Solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives, Applied Mathematics and Computation, 247, 213 – 217 (2014). https://doi.org/10.1016/j.amc.2014.08.101

66.

N. K. Vitanov, Z. I. Dimitrova, T. I. Ivanova: On solitary wave solutions of a class of nonlinear partial differential equations based on the function 1/coshn \((x+t)\)Aplied Mathematics and Computation, 315, 372 – 380 (2017). https://doi.org/10.1016/j.amc.2017.07.064

67.

N. K. Vitanov, Z. I. Dimitrova, K. N. Vitanov: Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications, Applied Mathematics and Computation, 269, 363 – 378 (2015). https://doi.org/10.1016/j.amc.2015.07.060

68.

N. K. Vitanov: Recent developments of the methodology of Modified Method of Simplest Equation with application, Pliska Studia Mathematica Bulgarica, 30, 29 – 42 (2019)

69.

N. K. Vitanov: Modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: History, recent developments of the methodology and studied classes of equations, Journal of Theoretical and Applied Mechanics, 49, 107 – 122 (2019). https://doi.org/10.7546/jtam.49.19.02.02

70.

N. K. Vitanov: The simple equations method (SEsM) for obtaining exact solutions of nonlinear PDEs: Opportunities connected to the exponential functions, AIP Conference Proceedings, 2159, 030038 (2019). https://doi.org/10.1063/1.5127503

71.

N. K. Vitanov, Z. I. Dimitrova: Simple equations method (SEsM) and other direct methods for obtaining exact solutions of nonlinear PDEs, AIP Conference Proceedings, 2159, 030039 (2019). https://doi.org/10.1063/1.5127504

72.

E. V. Nikolova, I. P. Jordanov, Z. I. Dimitrova, N. K. Vitanov: Evolution of nonlinear waves in a blood-filled artery with an aneurysm, AIP Conference Proceedings, 1895, 07002 (2017).https://doi.org/10.1063/1.5007391

73.

N. K. Vitanov, Z. I. Dimitrova: Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation, Journal of Theoretical and Applied Mechanics, Sofia, 48, No. 1, 59 – 68 (2018). https://doi.org/10.2478/jtam-2018-0005

74.

I. P. Jordanov, N. K. Vitanov: On the Exact Traveling Wave Solutions of a Hyperbolic Reaction-Diffusion Equation, Studies in Computational Intelligence, 793, 199-210 (2019). https://doi.org/10.1007/978-3-319-97277-0_16

75.

E. V. Nikolova, M. Chilikova-Lyubomirova, N. K. Vitanov: Exact solutions of a fifth-order Korteweg-de Vries-type equation modeling nonlinear long waves in several natural phenomena, AIP Conference Porceedings, 2321, 030026 (2021). https://doi.org/10.1063/5.0040089

76.

N. K. Vitanov: Simple equations method (SEsM) and its connection with the inverse scattering transform method, AIP Conference Porceedings, 2321, 030035 (2021). https://doi.org/10.1063/5.0040409

77.

E. V. Nikolova, D. Z. Serbezov, I. P. Jordanov, N. K. Vitanov: Non-linear Waves of Interacting Populations with DensityDependent Diffusion. Studies in Computational Intelligence, 961, Springer Nature Switzerland AG, 961, 324 - 332. https://doi.org/10.1007/978-3-030-71616-5_29

78.

N. K. Vitanov, Z. I. Dimitrova, K. N. Vitanov: Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods, Entropy, 23, 10 (2020). https://doi.org/10.3390/e23010010

79.

N. K. Vitanov. Schrödinger Equation and Nonlinear Waves, in Understanding the Schrödinger Equation edited by V. Simpao, H. Little, Nova Science Publishers, New York, pp. 37 - 92 (2020)

80.

N. K. Vitanov, Z. I. Dimitrova: Simple equations method (SEsM) and its particular cases: Hirota method, AIP Conference Proceeding, 2321, 030036 (2021). https://doi.org/10.1063/5.0040410

81.

Bialynicki-Birula Iwo, Mycielski Jerzy: Nonlinear wave mechanics, Annals of Physics, 100, Issue 1-2, pp 62 - 93, (1976).https://doi.org/10.1016/0003-4916(76)90057-9

82.

Giuliana Lauro, A note on a Korteweg fluid and the hydrodynamic form of the logarithmic Schrödinger equation, Geophysical & Astrophysical Fluid Dynamics, 102 (4), pp 373-380, (2008), Taylor & Francis. https://doi.org/10.1080/03091920801956957

83.

Rémi Carles, Isabelle Gallagher. Universal dynamics for the defocusing logarithmic Schrodinger equation. Duke Mathematical Journal, (2018), 167 (9), pp.1761-1801. https://doi.org/10.1215/00127094-2018-0006

Titel: An Application of the Simplest Equations Method to Logarithmic Schrödinger Equation
verfasst von: Ivan P. Jordanov
Verlag: Springer Nature Switzerland
Buch: New Trends in the Applications of Differential Equations in Sciences
Print ISBN: 978-3-031-53211-5

Electronic ISBN: 978-3-031-53212-2

Copyright-Jahr: 2024
DOI: https://doi.org/10.1007/978-3-031-53212-2_15

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