Abstract
In this paper, three Keller–Segel models are considered from the point of Lie symmetry analysis, conservation laws, symmetry reduction, and exact solutions. By means of Lie symmetry analysis, we first obtain all the symmetries for the three models. Based on the obtained symmetries, many non-trivial and explicit conservation laws for the three models are obtained with the help of Ibragimov’s new conservation theorem. Applying the characteristic equations of the obtained symmetries, symmetry reductions and exact solutions are obtained, including solutions expressed by rational functions and Bessel functions.
Highlights
The famous chemotaxis model was proposed by Keller and Segel in the 1970s to describe the aggregation of cellular slime molds Dictyostelium discoideum in response to the chemical cyclic adenosine monophosphate [1, 2]
6 Conclusions In summary, by performing Lie symmetry analysis on equations (1.2), (1.3), and (1.4), we obtain their Lie symmetries and find that their Lie symmetries are spanned by finite dimensional Lie algebra
According to the relationship between symmetry and conservation laws given by Ibragimov, many explicit and non-trivial conservation laws for the three equations are derived
Summary
The famous chemotaxis model was proposed by Keller and Segel in the 1970s to describe the aggregation of cellular slime molds Dictyostelium discoideum in response to the chemical cyclic adenosine monophosphate [1, 2]. Lie symmetry analysis has been extended to fractional partial differential equations (FPDEs) in recent years [22,23,24,25] This method has had a profound impact on both pure and applied areas of mathematics, physics, mechanics, etc. For the special case φ(v) = v, ε = 0, f (u, v) = βu – λv, Lie symmetry analysis and self-similar solutions are considered in [26], and a natural continuation of [26] for a (1 + 2)-dimensional Keller–Segel model is considered in [27, 28]. In terms of the Lie symmetry analysis method, we can obtain all of the geometric vector fields of the three systems. Substituting (3.5) and (3.7) into system (3.6), the adjoint system for system (1.2) is expressed as follows: Zt dzxx δZxvx βω ωt αωxx δZxxu δZxux λω (3.8)
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