One-dimensional parabolic-parabolic Keller-Segel (PP-KS) model of chemotaxis is considered. By using the generalized tanh function method, G'(/G)-expansion method and variable-separating method, plenty of new explicit exact solutions, including travelling wave solutions and non-travelling wave solutions, are obtained for the PP-KS model. Compared to the existing results, more new exact solutions are derived and the obtained solutions all have explicit expressions.
| Published in | Applied and Computational Mathematics (Volume 7, Issue 2) |
| DOI | 10.11648/j.acm.20180702.13 |
| Page(s) | 50-57 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Keller-Segel Model, Generalized Tanh Function Method, (G'/G)-Expansion Method, Variable-Separating Method, Exact Solutions
| [1] | R. Hirota and J. Satsuma, “Soliton solutions of a coupled Korteweg-de Vries equation,” Phys. Lett. A, 1981, 85, 407–408. |
| [2] | M. J. Ablowitz and H. Segur, “Soliton and the inverse scattering transformation,” SIAM, Philadelphia, PA, 1981. |
| [3] | M. Wadati, “Wave propagation in nonlinear lattice. I,” J. Phys. Soc. Jpn., 1975, 38, 673– 680. |
| [4] | R. Conte and M. Musette, “Painleve analysis and Backlund transformation in the Kuramoto-Sivashinsky equation,” J. Phys. A: Math. Gen., 1989, 22, 169–177. |
| [5] | L. H. Zhang, X. Q. Liu and C. L. Bai, “New multiple soliton-like and periodic solutions for (2+1)-dimensional canonical generalized KP equation with variable coefficients,” Commun. Theor. Phys., 2006, 46, 793–798. |
| [6] | C. L. Bai, C. J. Bai and H. Zhao, “A new generalized algebraic method and its application in nonlinear evolution equations with variable coefficients,” Z. Naturforsch. A, 2005, 60, 211–220. |
| [7] | G. W. Wang, “Symmetry analysis and rogue wave solutions for the (2+1)-dimensional nonlinear Schrödinger equation with variable coefficients,” Appl. Math. Lett., 2016, 56, 56–64. |
| [8] | H. Z. Liu and Y. X. Geng, “Symmetry reductions and exact solutions to the systems of carbon nanotubes conveying fluid,” J. Differential Equations, 2013, 254, 2289–2303. |
| [9] | E. G. Fan, “Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics,” Chaos, Solitons and Fractals, 2003, 16, 819–839. |
| [10] | D. S. Wang and H. B. Li, “Elliptic equation’s new solutions and their applications to two nonlinear partial differential equations,” Appl. Math. Comput., 2007, 188, 762–771. |
| [11] | E. F. Keller and L. A. Segel, “Initiation of slime mold aggregation viewed as an instability,” J. Theor. Biol., 1970, 26, 399–415. |
| [12] | E. F. Keller and L. A. Segel, “Model for chemotaxis,” J. Theor. Biol., 1971, 30, 225–234. |
| [13] | E. F. Keller and L. A. Segel, “Traveling bands of chemotactic bacteria: A theorectical analysis,” J. Theor. Biol., 1971, 26, 235–248. |
| [14] | J. Adler, “Chemotaxis in bacteria,” Science, 1966, 153, 708–716. |
| [15] | J. Adler, “Chemoreceptors in bacteria,” Science, 1969, 166, 1588–1597. |
| [16] | Z. A. Wang, “Mathematics of traveling waves in chemotaxis,” Discrete Cont. Dyn. B, 2013, 18, 601–641. |
| [17] | D. Horstmann, “From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II,” Jahresber. Deutsch. Math.-Verein., 2004, 106, 51–69. |
| [18] | T. Hillen and K. J. Painter, “A user's guide to PDE models for chemotaxis,” J. Math. Biol., 2009, 58, 183–217. |
| [19] | D. Horstmann, “From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, I,” Jahresber. Deutsch. Math.-Verein., 2003, 105, 103–165. |
| [20] | J. Murray, “Mathematical Biology: II. Spatial Models and Biomedical Applications,” 3rd edition, Springer, New York, 2003. |
| [21] | M. Shubina, “The 1D parabolic-parabolic Patlak-Keller-Segel model of chemotaxis: the particular integrable case and soliton solution,” J. Math. Phys., 2016, 57, 091501. |
| [22] | H. T. Chen and H. Q. Zhang, “New double periodic and multiple soliton solutions of the generalized (2+1)-dimensional Boussinesq equation,” Chaos Solitons and Fractals, 2004, 20, 765–769. |
| [23] | L. H. Zhang, L. H. Dong and L. M. Yan, “Construction of non-travelling wave solutions for the generalized variable-coefficient Gardner equation,” Appl. Math. Comput., 2008, 203, 784–791. |
| [24] | M. L. Wang, X. Z. Li and J. L. Zhang, “The (G '/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Phys. Lett. A, 2008, 372, 417–423. |
| [25] | G. W. Wang, T. Z. Xu, R. Abazari, Z. Jovanoski and A. Biswas, “Shock waves and other solutions to the Benjamin-Bona-Mahoney-Burgers equation with dual power law nonlinearity,” Acta. Phys. Pol. A, 2014, 126, 1221–1225. |
| [26] | R. X. Cai, Q. B. Liu, “A new method for deriving analytical solutions of partial differential equations algebraically explicit analytical solutions of two-buoyancy natural convection in porous media,” Sci. China. Ser. G, 2008, 51, 1733–1744. |
APA Style
Lihua Zhang, Lixin Ma, Fengsheng Xu. (2018). New Explicit Exact Solutions of the One-Dimensional Parabolic-Parabolic Keller-Segel Model. Applied and Computational Mathematics, 7(2), 50-57. https://doi.org/10.11648/j.acm.20180702.13
ACS Style
Lihua Zhang; Lixin Ma; Fengsheng Xu. New Explicit Exact Solutions of the One-Dimensional Parabolic-Parabolic Keller-Segel Model. Appl. Comput. Math. 2018, 7(2), 50-57. doi: 10.11648/j.acm.20180702.13
AMA Style
Lihua Zhang, Lixin Ma, Fengsheng Xu. New Explicit Exact Solutions of the One-Dimensional Parabolic-Parabolic Keller-Segel Model. Appl Comput Math. 2018;7(2):50-57. doi: 10.11648/j.acm.20180702.13
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Download@article{10.11648/j.acm.20180702.13,
author = {Lihua Zhang and Lixin Ma and Fengsheng Xu},
title = {New Explicit Exact Solutions of the One-Dimensional Parabolic-Parabolic Keller-Segel Model},
journal = {Applied and Computational Mathematics},
volume = {7},
number = {2},
pages = {50-57},
doi = {10.11648/j.acm.20180702.13},
url = {https://doi.org/10.11648/j.acm.20180702.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180702.13},
abstract = {One-dimensional parabolic-parabolic Keller-Segel (PP-KS) model of chemotaxis is considered. By using the generalized tanh function method, G'(/G)-expansion method and variable-separating method, plenty of new explicit exact solutions, including travelling wave solutions and non-travelling wave solutions, are obtained for the PP-KS model. Compared to the existing results, more new exact solutions are derived and the obtained solutions all have explicit expressions.},
year = {2018}
}
TY - JOUR T1 - New Explicit Exact Solutions of the One-Dimensional Parabolic-Parabolic Keller-Segel Model AU - Lihua Zhang AU - Lixin Ma AU - Fengsheng Xu Y1 - 2018/03/07 PY - 2018 N1 - https://doi.org/10.11648/j.acm.20180702.13 DO - 10.11648/j.acm.20180702.13 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 50 EP - 57 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20180702.13 AB - One-dimensional parabolic-parabolic Keller-Segel (PP-KS) model of chemotaxis is considered. By using the generalized tanh function method, G'(/G)-expansion method and variable-separating method, plenty of new explicit exact solutions, including travelling wave solutions and non-travelling wave solutions, are obtained for the PP-KS model. Compared to the existing results, more new exact solutions are derived and the obtained solutions all have explicit expressions. VL - 7 IS - 2 ER -
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