Volume 7, Issue 2, April 2018, Page: 50-57
New Explicit Exact Solutions of the One-Dimensional Parabolic-Parabolic Keller-Segel Model
Lihua Zhang, Department of Mathematics, Dezhou University, Dezhou, China
Lixin Ma, Department of Mathematics, Dezhou University, Dezhou, China
Fengsheng Xu, Department of Mathematics, Dezhou University, Dezhou, China
Received: Jan. 30, 2018;       Accepted: Feb. 11, 2018;       Published: Mar. 7, 2018
DOI: 10.11648/j.acm.20180702.13      View  1039      Downloads  88
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.
Keywords
Keller-Segel Model, Generalized Tanh Function Method, (G'/G)-Expansion Method, Variable-Separating Method, Exact Solutions
To cite this article
Lihua Zhang, Lixin Ma, Fengsheng Xu, New Explicit Exact Solutions of the One-Dimensional Parabolic-Parabolic Keller-Segel Model, Applied and Computational Mathematics. Vol. 7, No. 2, 2018, pp. 50-57. doi: 10.11648/j.acm.20180702.13
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[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.
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