Conservation laws, symmetry reductions, and exact solutions of some Keller–Segel models

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. MSC: 76M60; 58J70; 35L65

Invariant Solutions and Conservation Laws of the Variable-Coefficient Heisenberg Ferromagnetic Spin Chain Equation

“…where k i (i = 1, 2, 3, 4) are arbitrary constants. Substituting Equations (19) and 20into 3, we have…”

“…and A 1 , B 1 are free parameters. Based on Equations (19), (23), and (24), we obtain the following trigonometric function solutions for system (3)…”

“…On the one hand, based on the Lie group method, we can obtain new exact solutions directly or from the known ones or via similarity reductions; on the other hand, the conservation laws can be constructed via the corresponding Lie point symmetries. Recently, invariant solutions of a class of constant and variable coefficient NLPDEs have been obtained by virtue of this method, such as Keller-Segel models [19], generalized fifth-order non-linear integrable equation [20], KdV equation [21], and Davey-Stewartson equation [22].…”

Persistence of Traveling Waves to the Time Fractional Keller-Segel System With a Small Parameter

“…Inspired by the work based on Keller and Segel, the diverse dynamics for the original Keller-Segel system have become critical issues in the last decades [4,10,11,21,33,37,38,39,40,43]. It is seen that various versions to the Keller-Segel system are available, depending on different phenomena and scales.…”

Symmetry reductions of a (2 + 1)‐dimensional Keller–Segel model

“…In particular, they constructed travelling wave and self‐similar solutions for this system. Furthermore, in Zhang and Xu, 8 three KS models in (1 + 1) dimensions were considered from the point of Lie symmetry analysis and conservation laws. Based on the admitted symmetries, the authors obtained conservation laws by applying Ibragimov's conservation theorem.…”